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Also any Cayley algebra is simple (hence central simple over F); this is obvious for Cayley division algebras, but not for the split Cayley algebra. If 8 is any nonzero ideal of 6 = 0 + uQ, there is x = a + ub # 0 in 23 (a, b in Q). If a # 0, then 1 E QaQ E QxQ + Q(ub)Q c 23 + uQ, 4. Cayley Algebras 49 implying 1 - vh is in 23 for some h in Q. Now 3 = F1 + Fs with sz = s u l , so s - pi = -p1 + (1 + /3) s # 0 for all /Iin F. Then 23 contains (1 - vh)s [v(sh)](l - vh) = s - p ( h ) i = q # 0 + + in Q, and 1 E QqQ E 23, so 23 = 6 in this case also.

Hence D is in f? if and only if D = R, + L, . If 21 contains a left identity e, then L , = 1, and 0 = LeD = L e x + y e= L, + L e L y= L, + L , , so that D = R, - L, (which is a derivation of % for every x in a). Guided by the known structure of finite-dimensional associative algebras (of arbitrary characteristic) and Lie algebras of characteristic 0, II. , 1 ) . It is easy to see that the Gi are uniquely determined by 2l. (%) generated by 6. Hence, if a finitedimensional algebra 2I is % = 6, @ ...

6. , t ) is a subspace of 24, and Gij E Cp, the set of properly nilpotent elements of 2l. Proof. Since 2100 is a nilalgebra, we have seen above that Goo= 2100c Cp. Also we have Go,= 210j ( j = 1, ... 30). $3 ( i = 1, ... 30). Thus we need only consider G i jwith i # 0 , j # 0. Suppose that s,,, sij are in G i j ; then sijajiand sljaj, in a,, belong to the ideal of nilpotent elements of aii,and (asij + /ls;i)ajiis nilpotent for all a, p in F. That is, asfj psij is in G,,, and G,j is a subspace of '$I.

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