A useful proposition for division algebras of small degree

Abstract:

Let $F$ be a field and let $D$ be an $F$-central division algebra of degree $n$. We present a short, elementary proof of the following statement: There is an $n - 1$-dimensional $F$-subspace $V$ of $D$ such that for every nonzero element $\nu$ of $V$, ${\text {Tr}}(\nu ) = {\text {Tr(}}{\nu ^{ - 1}}) = 0$. We then indicate how one can use this result to obtain the basic structural results on division algebras of degree three and four (results of Wedderburn and Albert, respectively).