5) See Theorem 2 of PAR.The existence of the characteristic root one-half will cause our formulas to contain a number of fractions such as 1/2, 3/2, and 5/2.The reader should always read such formulas as 1/2 xyz as {l/2)xyz and not as (2xyz)~l.(') See Theorem 6 of JA2. 1950] POWER-ASSOCIATIVE COMMUTATIVE ALGEBRAS 505(3) ai/2*i = ai/nSi/i(xi) + ßi/2^o(*i).Here ai/2Si/2(xi) is in 2L(l/2) so that Si/2(xi) is an endomorphism of 2Iu(l/2).Also ßi/25o(xi) is in 3L(0) and so 5o(xi) is a homomorphism of 3I"(l/2) into 3L(0).Note that when 21 is a linear algebra over a field % the mapping •Si/2(xi) is a linear transformation of 2L(l/2), and So(xi) is a linear mapping of
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A. A. Albert (1950) studied this question.