In my Lemma 4,1 have in fact reduced the condition that A be a division algebra from a condition that a quartic form in sixteen variables be not a null form to an equivalent condition on a quadratic form in only six variables. It is the application of this far simpler condition that has enabled me to prove the existence of non-cyclic algebras. I have shown in the above that among the algebras considered by Brauer there exist non-cyclic division algebras and also algebras not division algebras.

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... here remains the question as to whether any of the algebras of Brauer are cyclic division algebras. I have recently proved* that the algebra A =BXC over R (u, v), where we replace u by -2u s > take a to be a rational number which is a sum of two squares and not a square, and take b= -1, is a cyclic normal division algebra. This is one of the algebras of Brauer when we pass to a new basis of B by taking i to be replaced by u~H whose square is -2w, and then replace u by the equivalent indeterminate -2u.