The structure of alternative division rings

Universal and homogeneous embeddings of dual polar spaces of rank 3 defined over quadratic alternative division algebras

“…Explicit constructions of such alternative division rings can be found in Jacobson [17, p. 426 The multiplication in a Cayley-Dickson division algebra is not associative. In fact, it is a result due to Bruck and Kleinfeld [3] and Kleinfeld [19] that the Cayley-Dickson division algebras are the only alternative division rings in which the multiplication is not associative. A proof of that result can also be found in Tits and Weiss [28,Chapter 20] and Van Maldeghem [29,Appendix B].…”

“…. , 13} defined by (0, 13)(1, 7)(2, 8) (3,9). Then the point of PG(7 + 6n, K) with all coordinates 0 except for the one in the i -th place which is equal to 1 does not belong to p ζ , showing that p does not belong to R.…”

Compact totally disconnected Moufang buildings

“…The first assertion was first proved in [5] and [21]; see [37, 20.2-20.3] for another proof. Therefore k is infinite by, for example, [37, 9.9(v)].…”

Representations of Alternative Algebras

“…If the characteristic of F is not two, then the identities (3) define an alternative algebra. Let 23 be a vector space over F. Following Eilenberg [3, §2] for all x, z in SI, v in 23, where the associator is defined as in (2) except that one argument is in 23. The identities (5) are equivalent to the assumption that every associator with one argument in 23 and two in 31 "alternates.…”