QN THE WEDDERBURN PRENQPLE THEOREM FOR COMMUTATNE POWER -ASSOC!AT2VE ALGEBRAS Though For fin Diagram: 9f Ph. D. MICHIGME 3TH? UNEVERSITY Robert Louis Hemminger E953 This is to certify that the thesis entitled On the Wedderburn Principle Theorem for Commutative Power-Associative Algebras. presented by Robert Louis Hemminger has been accepted towards fulfillment of the requirements for (ITEZOALLitlA Qhfilmmijal Major professor [hm June 13, 1963 0-169 LIBRARY Michigan State University MSU RETURNING MATERIALS: Place in book drop to ”saunas remove this checkout from n your record. FINES will be charged if book is returned after the date stamped below. ABSTRACT ON THE WEDDERBURN PRINCIPLE THEOREM FOR COMMUTATIVE POWER-ASSOCIATIVE ALGEBRAS by Robert Louis Hemminger let A be a strictly power—associative algebra with radical N and such that the difference algebra A - N is separable. Then we say that A has a Wedderburn de- composition if A has a subalgebra 8‘: A - N with A = S + N (vector space direct sum). The so—called Wedderburn Principle Theorem for associative algebras can be stated as follows: If A - N is separable for an associative algebra A then A has a Wedderburn decomposition. The analogue of this theorem for alternative and Jordan algebras has also been proved. This thesis investigates this theorem for the commutative strictly power-associative algebras. Our first result of primary importance is an example of a commutative power-associative algebra which does not have a Wedderburn decomposition. Since the base field in this example only has the restriction that it have char- acteristic not 2, 3, 5 we cannot even hope to prove the Wedderburn Principle Theorem.for commutative strictly power-associative algebras by only restricting the base field. On the other hand we show that large classes of commutative strictly power-associative algebras do have 1 Robert Iouis Hemminger Wedderburn decompositions by proving the following two ‘theorems. (a) If A is a commutative strictly power-associative algebra of characteristic not 2 such that A - N'= B1 ED ... @Bt is separable such that each B1 is simple and has three pairwise orthogonal idempotents then A has a Wedderburn decomposition. (b) Let T be the class of commutative strictly power-associative algebras of characteristic not 2 that satisfy a property P such that A in p implies that every subalgebra of A is in p. Then every algebra in $ has a Wedderburn decomposition if and only if every algebra in N that has at most two pairwise orthogonal idempotents has a Wedderburn decomposition. This last result is used to show that every stable commutative power—associative algebra over an algebraically closed field F of characteristic zero has a Wedderburn decomposition. In the associative, alternative, and Jordan cases the proof was accomplished in two stages; namely, N2 = 0 and N2 % O. For N2 = 0 an actually Wedderburn de- composition was constructed while for N2 # O a nil ideal M with O C M C N was constructed in terms of N and a Wedderburn decomposition was established by a simple induction argument. Robert Louis Hemminger In our case we didn't encounter the case N2 = 0 but our proofs did bear some resemblance to the case N2 e o. This similarity is reflected in the following result which was our basic tool in establishing (a) and (b) above. If M is any ideal of A with M e o, N, A then A has a Wedderburn decomposition. Using this result repeatedly for various ideals we were able to re- duce A sufficiently to be able to construct a Wedderburn decomposition for it. ON THE WEDDERBURN PRINCIPLE THEOREM FOR COMMUTATIVE POWER-ASSOCIATIVE ALGEBRAS By Robert Louis Hemminger A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mathematics 1963 329067 4/2 4/4. 4 ACKNOWLEDGEMENTS I am indebted to Professor R. H. Oehmke for suggesting this thesis problem and for his helpful guidance in completing it. I especially wish to express my deep gratitude for his kind consideration and encouragement throughout my stay at Michigan State University. This thesis was written while I held a fellowship from the Institute of Science and Technology at Ann Arbor, Michigan. 11 Dedicated to Azora iii 2. 3. 1+. 5. CONTENTS IntrOdUCtionooo00000000000000.0000... mampleOOOOOO......OOOOOOOOOOOOOOOOOO Pairwise orthogonal idempotents...... Classes of algebras with Wedderburn decompOSj-tionSO......OOOOOOOOOOOOOOOO Proof of Theorem 2................... a. Preliminaries.................. b. Completion of the proof........ A reduCtion theoremoooéooooooooooocoo An application.0.000000000000000.coo. Bibliography.OOOOOOOOOOOOOOOOOOOOOOOO iv Page 12 18 18 22 31 3h #3 1. Introduction Let A be a strictly power-associative algebra with radical N and such that the difference algebra A — N is separable. Then we say that A has a Wedderburn decomposition if A has a subalgebra S 3’A — N' with A = S + N (vector space direct sum). As a matter of terminology, by an algebra we shall always mean a finite dimensional vector space on which there is a multiplication defined which satisfies both distributive laws. The radical of a strictly power- associative algebra is the unique maximal nil ideal and a non-nil algebra with zero radical is said to be semi- simple. A simple algebra is a non-nil algebra with no proper ideals. An algebra A is power-associative if xaxB = xa+B for all positive integers a and B, and every x in A. An algebra A over a base field F is strictly power-associative if xa’xB = xa+B for all positive integers a and B, and every x in AK where K is any scalar extension of F. The characteristic of an algebra is the characteristic of its base field. If the characteristic is not 2, 3, or 5 then strict power—associativity is equivalent to power-associativity [7, pp. 36h]. An algebra is separable if it is semi- simple over every scalar extension of the base field. The elements of the difference algebra A - N are the classes [a], defined for every a in A, where [a] = [b] if and only if a — b is in N, [a] + [b] = [a + b], and [allb] = [ab]. The basic structure theory of commutative power- associative algebras of characteristic not 2, 3, or 5 was given by Albert in [A]. Most of these results were carried over to commutative strictly power-associative algebras of characteristics 3 and 5 by Kokoris in [7]. Any reference to [A] will thus be understood to imply a reference to the corresponding result in [7]. Most of the results on commutative strictly power- associative algebras depend on an idempotent decomposition where an element e in A is idempotent if e2 = e # O. For the idempotent e we have A = Ae(1) + Ae(1/2) + Ae(0) where x is in Ae(l) if and only if ex = Xx for x = O, 1/2, 1. Moreover Ae(1) and Ae(O) are orthogonal subalgebras of A and for x = O, 1 we have Ae(x)Ae(1/2) ; Ae(1/2) + Ae(1 - x) and Ae(1/2)Ae(1/2) g Ae(1) + Ae(o) (the product BC of two subspaces B and C of the algebra A is the set of all finite sums Ebc, b in B and c in C; in particular B2 = BB and Bm = BB‘m"1 for m_2.2). For x in A we will frequently use this idempotent decomposition of A to express x uniquely in the form x = x1 + Xlfi3+ xO where x)V for x = O, 1/2, 1. Every semi-simple commutative strictly is in Ae(x) power-associative algebra of characteristic not 2 has a unity element and can be expressed uniquely as a direct sum of simple algebras. These results are all contained in [h]. The characterization of the simple, and hence semi- simple, commutative strictly power-associative algebras is now essentially complete [see'Hfl so it is desirable to see if a Wedderburn decomposition can be given for them. The example in §2 shows that this is not possible in general. The purpose of this thesis is to show that a large class of the commutative strictly power-associative algebras do have Wedderburn decompositions and to point out what one might expect in those that do not have a Wedderburn decomposition. In §h we show that if A is a commutative power- associative algebra with characteristic not 2, A - N is separable, and A - N = B1 @B2 (-9 ... @Bt where each B1 is simple and contains three pairwise orthogonal idempotents then A has a Wedderburn decomposition. In §6 we show that if T is the class of commutative strictly power-associative algebras having a property P then every algebra in T has a Wedderburn decomposition if and only if every algebra in T having at most two pairwise orthogonal idempotents has a Wedderburn decomposition. This result is applied in §7 to the class of stable algebras over algebraically closed fields of characteristic zero. The so—called Wedderburn Principle Theorem for associ— ative algebras can be stated as follows: If A - N is separable for an associative algebra A then A has a Wedderburn decomposition. A proof of this can be found in [1, Theorem 23, pp. #7]. This theorem was generalized to alternative algebras by Schafer [12] and its analogue for Jordan algebras was proved by Penico [11]. Previous to that Albert had proved it for an important class of Jordan algebras [2]. In all of these cases the method was basically the following. For N2 a O a subalgebra isomorphic to A - N‘ was actually constructed and for N2 i O a nil ideal M # O, N was constructed in terms of N‘ and the theorem obtained by the induction argument we have given for the proof of Lemma 2.1. In each case the construction of M depended on knowing that an ideal is nilpotent if and only if it is nil (M is nilpotent if Mn = O for some positive integer n while M is nil if each element of M is nilpotent, that is, for each x in M there is a positive integer n, depending on x, such that xn = 0). But it is unknown if this is the case in commutative strictly power-associative algebras or not. This difficulty is mainly circumvented by lemma 2.2 for according to that result if M is any ideal of A we can assume M = O, N, or A. By repeated use of lemma 2.2 we are able to reduce A sufficiently to actually construct a Wedderburn decomposition for it. This is done in the proof of Theorem 2. Since the latter part of this proof requires some preliminary material and is quite long we have put it in a separate section. We will always let N represent the radical of the algebra A and we assume N'# O, A since otherwise A has a trivial Wedderburn decomposition. Unless otherwise specified we will understand that the generic symbol A represents a commutative strictly power-associative algebra of characteristic not two with A - N separable. 2. Example Let A be the 6-dimensional commutative algebra with basis e11, e12, e21, e22, m, n and multiplication table 62 "' e e2 11 ‘ 11’ 22 922’ e11e12 = e22612 a 1/2e12’ e11e21 = e22821 = 1/2821’ e11n = e12m = n’ eaem = e21n = m, = 1/2(e11 + e + m + n), and all other products e12e21 zero. 22 The algebra A is commutative by definition. If we restrict A to have a base field F of characteristic not 2, 3, or 5 and let x be a general element of A (expressed in terms of the basis elements) then by compu— tation we find that x2x2 = (xex)x. So by [3, lemma A, pp. 55h] A is power-associative. For an algebra B of characteristic not 2, Bi is the algebra with the same additive group as B but the multiplication of Bi is defined by xy = 1/2(x o y + y o x) where x o y is the product of x and y in B. The radical N of A is spanned by m and n, N2 = O, and in the notation of the last paragraph we see that A - N': F: with basis [e11], [e12], [e21], [e22] where F2 is the algebra of all 2 by 2 matrices over F. Suppose A had a subalgebra S g A - N. Then 8 would have the usual matrix basis g11, g12, g21, g22 + for F2 and there would be an automorphism o of A — N such that o([e ]. But this is a change of 13]) = [$13 basis for the 2 by 2 matrices so there is a nonsingular element [y]= d[e11] + ele121+ 7Ie21] + olegel in A - N, with A = as - B7 # 0, such that [gij] = [y] 0 [e13] o [y]-1 (note that this multiplication takes place in F2). But [y].1 = A-1(o[e11] - Ble12] - 7le21] + ale so 22]) .1 o [yl-l we have computing IgiJ] = [y] o [eiJ -1 g11 = A (doe1 - dee12 + 78e2 - B7e22 + 61m + egn) 1 1 _ —1 2 _ 2 g12 ‘ A (’ “7811 + “ e12 7 e21 + “7822 + 91m + 92“) g = A—1(a8e — B2e + 82e — Boe + x m + X n) 21 11 12 21 22 1 2 -1 g22 _ A (- B7e11 + doe12 - 75e21 + doe22 + r1m + w2n) Equating coefficients of m and n in the products (for example 1/2 8;12 yields equations in gijgki g11g12 and which force A = O. has no subalgebra S g A — N and hence the coefficients of m and n in are equal since gHg12 = 1/2 g12) a, 5: 7: 5: €19 €21 00-: “1: “2 But this is a contradiction so A A has no Wedderburn decomposition. This example of Wedderburn Principle commutative strictly over it shows we can restricting the base course shows we can not prove the Theorem for the class of all power—associative algebras. More— not even hope to prove it by only field for in our example the base field is arbitrary other than the restriction that the characteristic not be 2, 3, or 5. An algebra is called stable with respect to an idempotent e if Ae(x)Ae(1/2)(; Ae(1/2) for i = o, 1 and it is called stable if it is stable with respect to each of its idempotents. From the multiplication table for A above ne = m so A is not stable with respect to e 21 11 (or e22). Now by Theorem 2 of [9, pp. 698] if f is any other idempotent of A (and f # 1 = 211 + e22) then f = 1/2(1 + w) where w2 = 1 and w = u(e - e22) + w1 + w + w2 where W12 # O is 11 2 1 in A (1/2) and w is in A (A) n N for 811 X e11 X a O, 1. Computing the general element w with these properties we find that A is not stable with respect to f = 1/2(1 + w) either. That is, our example is not stable with respect to any idempotent. Looking at the other side of the coin, we show in §7 that A has a Wedderburn decomposition if it is stable over an algebraically closed base field of characteristic zero. 3. Pairwise orthogonal idempotents In this section we will assume the algebra A has the element 1 as a unity. Based upon and related to the decomposition of A by a single idempotent Albert has given in IL, §5l a decomposition of A relative to a set of pairwise orthogonal idempotents e1, e2, ..., et for which 1 = e + e + ... + e It is shown that we 1 2 t' can write A in a vector space direct sum A =2 A13 153 for i, j = 1, 2, ..., t where Aii a Aei(1) and A = A 13 31 Ae.(1/2) fl AeJ(1/2) When 1 ¥ J. Moreover i if g = ei + e.j for i # J then g is an idempotent = All + A13 + A”, 1130/2) =Zkgéi,J(Aik 1' AJk)’ and A (O)==§: ‘ Akfl’ We will have occasion to use 95 kzgéi J with A 1 gm these only for t = 3. In that case A = A11 + A22 + A33 + A12 + A13 + A23 and with i, j k distinct and g = e1 we have Ag(1) = All + A13 + AJJ’ eJ Ag(1/2) = A1k+ AJk, and Ag(o) = distinct we have Akk' For i, J, k, E 2 Aiig Aii AiiAiJ ; AM + AJ.J AliAJJ = AijAkZ = A1 iA k3 = 0 KW ID 1.4. 5.. AiJAjk AEJQA +A Since these relations are basic to much of our work we will generally use them without specific reference. Also related to pairwise orthogonal idempotents we have the following lemma. lemma 1: Let [u1], [u2], ..., [ut] be pairwise orthogonal idempotents in A - M, M a nil ideal of A, and let u = u1 + 112 + ... + ut. Then there exists an idempotent e and pairwise orthogonal idempotents ill. Al 10 e1, e2, ..., et in Ae(1) such that e = e1 + e2 + ... + e [e] = [u], and [e1] = [ui] for i = 1, 2, ..., t. More- t, over if A has 1 as a unity element and [1] = [u] then e = 1. P3993: The proof of the first part of the lemma is by induction and the case t = 1 is Lemma 1 of [2, pp. 1]. Here u = u,. Now [u]k = [uk] = [u] so u cannot be nilpotent. Hence the associative algebra of all poly- nomials in u, denoted by Flu], is not nilpotent and thus contains an idempotent e a f(u) for f in FIu]. Then [e] = [f(u)] = alu] where a = f(1) is in F. Thus dIu] = [e] = [e]2 = d2lu12 = aglul. Since e is an idempotent it is not in M so alu] # 0, d = 1, and [u] = [e] as desired. let w a u1 + u2 for t 2.2- Then u a w + u3 + ... + ut for pairwise orthogonal idempotents [w], [u3], ..., [ut]. By the induction hypothesis there exists an idempotent e and pairwise orthogonal idempotents f, e3, ..., et in Ae(1) such that e = f + e3 + ... + et, [el = [u], [f] = [w], and [e1] = [ui] for i = 3, ..., t. In particular f is an idempotent of A such that and u are 1 2 orthogonal idempotents (note that this is essentially [f] = [w] with w = u1 + u2, where u the case t = 2 only in that case f would have been obtained from the case t = 1 rather than from the 11 induction hypothesis). Then [f][u1l = [wllw — u2] = [w] - [uel = [u1]. If [fllx] a [x] for x in A we can write x a x + x1/2 + x and have 1 0 [x1] + [x1/2] + [x0] = [x] = [fllx] = [x1]+ 1/2[x1/2] so [x] = [x1]. Now [f][u1] = [u1] so there is an element x in Af(1) such that [x1] = [u1]. More- 1 over x is not nilpotent since [u1] isn't. Hence the 1 associative algebra F[x1] is not nilpotent and thus contains an idempotent e which is in Af(1) since 1 Af(1) is a subalgebra. Then Just as in the case t = 1 we have [e1] = [x1] and so [e1] = [u1]. Now e2 = f - e1 is an idempotent in Af(1), e (f - e1)e1 a e 2e1 - e = 0, [e2] a [f - e1] = If] - [e1] = 1 1 [w] - [u1] = [u2], and since e1 and e2 are in Af(1) they are orthogonal to ei for i = 3, ..., t. Thus e = f + e3 + ... + et a 21 the el are pairwise orthogonal idempotents with + e2 + e3 + ... + et where [e] = [u] and lei] = [uil for i a 1, 2, ..., t. Since f is in Ae(1) we have Af(1)gAe(1) so e1 and e2 are also in Ae(1) which completes the proof of the first part of the lemma. For 1 in A, 1 - (e + e2 + + e is either 1 t) zero or an idempotent of A. But [1] = [u1] + ... + [ut] = [e1] + ... + let] means that 1 - (e1 + ... + et) is in M, so it is nilpotent. Hence it is zero and 12 1=e1 As a consequence of Lemma 1 we immediately have 'l" 82 + co. 'l" et as deSiredo Corollary 1. Corollary 1: If M is a nil ideal of A then A has t pairwise orthogonal idempotents if and only if A - M has t pairwise orthogonal idempotents. h. Classes of algebras with Wedderburn decompositions Let a be the class of all commutative strictly power-associative algebras A that have a Wedderburn decomposition and for which A — N is simple. We are using B + C to mean the vector space direct sum of the subspaces B and C. In particular this means that B n C = 0. If in addition B and C are sub~ algebras of A such that BC = 0 then we write l3C)Ch This is called the direct sum of the subalgebras B and C. Theorem 1: let A be a commutative strictly power- associative algebra of characteristic not two. Then it is known that A - N = B16) ... @Bt where Bi is simple and has a unity element [ui]. Let ei be as in lemma 1. Then A has a Wedderburn decomposition if Ae (1) is in $1 for i =1, 2, .00, to 1 Proof: The proof is by induction on t. Let 13 e = e1 + e2 A1 = Ae1(1), A12 = Ae1(1/2)’ and A2 = Ae1(0)° Also = N n A1 for + ... + et as in lemma 1 and let and N let R i be the radical of A1 Remark: When B is a subspace of A then B - N is the subspace of A - N consisting of all classes [b] for b in B. When B is a subalgebra of A then B - N is a subalgebra of A - N and is isomorphic to B - Nb where N5 = N n B. From the above we clearly have A1 - N': B1, N A12 - N = O, and A2 " N: B26)... @Bto SO A12 ; No Let M = R1 + A12 + R By the definitions above R 2' i is a nil ideal of A and NQ M so AM = (A1+ A12 + A2) 1 (R1+ A + R2) g A1R1+ A1A12 + A12M + A2A12 + A232 _C_ M 12 and M is an ideal of A. Moreover if x is in M then x = a + n + b for a 6 R1, n e N, and b 6 R2. 2 2 So x is in a + b2 + N, x3 is in a3 + b3 + N, and by induction, xk is in ak + bk 4- N for every positive integer k. But a and b are nilpotent elements so for k sufficiently large we have xk in N so xk is nilpotent, x is nilpotent, and M is a nil ideal of A. Thus MQN so M=N and Ri=Ni 2-N=O so A12+A2QN and since A1 is in m it has a Wedderburn decomposition, for 1:31:20 Now for t = 1, A 1h say A1 = S1 + N1. Then A = S1 + N' is a Wedderburn decomposition for A. m N If t>1 then A2—N=A2-N2=B2®...®Bt where [ui] = [e1] is the unity element of Bi for i = 2, ..., t. Moreover (A2)e (1) = A8 (1) is in m i i so by the induction hypothesis A2 has a Wedderburn decomposition, say A2 = 82 + Né. Then A = (S1<:>S2) + N is a Wedderburn decomposition of A. Before we can put much confidence in the value of Theorem 1 we must at least know that the class N is of sufficient size to have some importance. That is the purpose of the next two theorems. Theorem 2: Let A be a commutative strictly power- associative algebra with a unity element and of charac— teristic not 2 such that A has three pairwise orthogonal idempotents and A - N is simple. Then A has a Wedderburn decomposition. .Prggf: The proof is by induction on n, the dimension of A. Then n.2 3 since A has three pairwise orthogonal idempotents. The theorem is trivial if n = 3 so assume every algebra of dimension less than n and of the type described in the theorem has a Wedderburn decomposition. Remark: For a nil ideal M of A, A - M is 15 semi-simple if and only if M = N. For M(; N since N is the maximal nil ideal of A. Thus N - M is a nil ideal of A - M. So for A - M semi-simple, N — M = O and N = M. Conversely if M‘ is a nil ideal of A - N then there is an ideal M in A such that Mg; M and M - N'= M'. But if b is in M then [b]S = O for some positive integer s so b d“ o is in N and (bs)t = O for some positive integer Thus M is a nil ideal of A so M = N and M' = O as desired. The major tool in the proof of Theorem 2 is the important Lemma 2.2. We first prove a special case of it. lemma 2.1: If M is a nil ideal of A with M # o, N then A has a Wedderburn decomposition. .nggf: For convenience we will write d(B) for the dimension of a subspace B. By the homomorphism theorems A - N": (A — M) - (N - M) so by the Remark above N - M is the radical of A - M. Now (A — M) - (N - M) is simple since A - N is simple and A — M has a unity element since A has one. And by Corollary 1, A - M has three pairwise orthogonal idempotents. But M is a proper ideal of A and we have 3 g d(A - M)‘< n so by the induction hypothesis A - M has a subalgebra CO such that 16 CO 3 (A - M) — (N - M) g A - N. Again by the homomorphism theorems A has a subalgebra C1 # O, A such that . 2 _ MC C1 (that 18 Mgc1 and Mgé C1) and CO— 01 M. Thus we have a proper subalgebra C1 of A such that C1 - MI: A - N. Similar to the considerations for A - M above we see that M is the radical of C1, C - M is 1 simple, C1 has three pairwise orthogonal idempotents, and 3 S_d(C1) < n. So by the induction hypothesis C1 C1 - M. Thus C is a subalgebra A - N. But C n N is a nil ideal “2 has a subalgebra C u? of A such that C of C so C n N'= 0 since C 3 A - N' which is simple. Therefore C + N is a subspace of A with d(C + N) = d(C) + d(N) = d(A - N) + d(N) = d(A). So A = c + N and this is a Wedderburn decomposition for A. lemma 2.2: If M is any ideal of A with M % O, N, or A then A has a Wedderburn decomposition. _P_I_‘_o_o_f_: By lemma 2.1 we can assume Ml N and NZ M since A - N is simple. If M n N 74 0 then it is a nil ideal of A different from O and N so by Lemma 2.1 A has a Wedderburn decomposition. If M n N = 0 then M1+ N is an ideal of A and (M'+ N) - N is a non—zero ideal of A - N. But A - N is simple so A = M«+ N with M n N’= O. This is a Wedderburn decomposition of A and completes the proof of lemma 2.2. 17 The remainder of the proof of Theorem 2 involves repeated applications of Lemma 2.2 to various ideals of A. By this method we are able to reduce the algebra A to one for which we can construct a Wedderburn decomposition. This is done in the next section. But for the moment let us assume that Theorem 2 is proved. We can then prove a more general result. Theorem 3: Let A be a commutative strictly power- associative algebra of characteristic not two such that, in the canonical representation, A - N = B1 6-) ... ®Bt where each Bi has three pairwise orthogonal idempotents. Then A has a Wedderburn decomposition. 33993: let [ui] be the unity element for Bi and let e1 be the pairwise orthogonal idempotents in A as in Lemma 1. Fix 1 and let f = e1. Then just as was done for 21 Af(1) - N 2’ 131 where Nf = N n Af(1) is the radical of in Theorem 1 we have Af(1) - Nf 2 Af(1). But f is the unity element for Af(1) and by Corollary 1 Af(1) has three pairwise orthogonal idempotents. So by Theorem 2, Af(1) is in m. But this is true for each i = 1, 2, ..., t so by Theorem 1 A has a Wedderburn decomposition. 18 5. Proof of Theorem 2 a. Preliminaries Since A is strictly power-associative we have x2x2 = (x2x)x for each x in A. The linearization of this identity gives h[(xy)(ZW) + (XZ)(YW) + (XW)(YZ)] (1) = x[y(zw) + z(wy) + w(yz)l + y[x(zw) + z(wx) + w(xz)] + z[x(yw) + y(wx) + w(xy)] + w[x(yz) + y(zx) + z(xy)]. We will also make use of some of the results of Albert on commutative strictly power—associative algebras; namely, results (5) and (8) of IA, Pp. 505-506]. We state them as (2) [W1/2(X1y1)]1/2 a [(W1/2X1)y1 + (W1/2y1)x1]1/2 (3) [W1/2(X1y1)]o = 2[(W1/2X1)y1+ (W1/2y1)x1]o (h) [(W1/2y1)xo]1 = 1/2[(W1/2Xo)y1]1 where 2%, A = O, 1/2, 1, is the Ae(x) component of z; e an idempotent. Before continuing we need to explain some new notation we will use. We have already commented on the vector space direct sum B + C. In the remainder of this 19 thesis we will IN) longer require that B + C indicate the direct sum of vector spaces; only the sum. That is we will now use B + C on occasions where B n C # 0. But in §7 we will sometimes want to explicitly indicate that we are using the vector space direct sum. In that case we will write B 4 C. We have also used the product BC previously. But it is too restrictive for our purposes now so we intro- duce a new product, B o C, of the subspaces B and G. Since A has a unity element, denoted by 1, and three pairwise orthogonal idempotents we can write 1 = e1 + e2 + e3 where the ei are pairwise orthogonal idempotents. Then as in §3 A has a corresponding de— composition as A =Zi