I was reading book of Humphreys on Lie algebra. In the first chapter, he introduced four classical algebra, and I thought, there was missing special unitary algebra. Can one give definition of it considering field $F$ to be of any characteristic?
I thought the following: let $F$ be a field with an automorphism $\sigma$ of order $2$ and let $(\cdot,\cdot)$ be a sesquilinear form on a vector space $V$ over $F$ (so it is linear in first coordinate, and $(x,ay)=\sigma(a)(x,y)$.)
Then unitary algebra may be the set of those $g\in \mathfrak{gl}(V)$ such that $(gx,y)=-(x,gy)$ for all $x,y\in V$. Am I right?
The special unitary algebra is subalgebra of unitary with trace $0$ elements.