Isotopes of some nonassociative algebras with involution

Tensor Products of Composition Algebras, Albert Forms and Some Exceptional Simple Lie Algebras

“…Suppose first that A is simple. Then, (mi,m2) = (4,1), (4,4), (8,1), (8,4), or (8,8). Now dim(C+(3i', Q)) = 2d_1, and so dim((LyLy)í) = 2a"1 or 2d"2 accord- PROOF.…”

“…The equivalence of (a) and (b) is proved in [8,Proposition 12.3]. Suppose then that (sé,-) and (sé1,-) are central division algebras and that there exists a Lie algebra isomorphism ep: 3T(sé,-) >-> .5í(sé', -).…”

“…By (6.3), the pure norm of any of these algebras represents 1, and so the Albert form of the tensor product of any two of them is isotropic. Also the unique quadratic extension 4(\f^l) of 4 is algebraically closed [21, Theorem VII.2.5], and so the Albert form of any twisted (8,8)-product algebra is isotropic. Thus, there is one (8, l)-product division algebra (( -1, -1, -1), -) and no division algebra forms of (8,m)-product algebras for m > 1.…”

“…THEOREM 7.13. The division algebra forms of (8,8)-product algebras over 43 are up to isotopy the algebras (7.14) ((pïi,oTa,TT3),-)®((-l,-l,e),-)> where p,o,T £ {-1,1} and (7.15) £ G {-1, -pTi, -oT2, -rT3,paTxT2,prTxT3,orT2T3, -porTxT2T3}.…”

“…Suppose first that (sé, -) is a division algebra form of an (8,8)-product algebra over 43 with Albert form Q. By Theorem 7.3, (sé,-) is an (8,8)-product division algebra. Hence, we may assume that (sé, -) = (gi, -)®(5%2,-), where gi, g7; are in the last column of Table 7.12.…”