(special) unitary Lie algebra

Let $V$ be a vector space over a field $K$ admitting an involution $\sigma :K\to K$, and let $B:V\times V\to 𝔽$ be a http://planetmath.org/node/SesquilinearFormsOverGeneralFieldshermitian form relative to $\sigma$. Then the unitary Lie algebra $𝔲(V,B)$, or just $𝔲(V)$, consists of the linear transformations $T$ satisfying

$$B(Tx,y)+B(x,Ty)=0,$$

for all $x,y\in V$. This is a Lie algebra over $k=\{\alpha \in K\mid {\alpha }^{\sigma }=\alpha \}$, but not over $K$ in the case that $K\ne k$ (because $B$ is linear in the first, but not in the second variable).

The special unitary Lie algebra $𝔰u(V,B)$, or just $𝔰u(V)$, consists of those linear transformations in $𝔲(V,B)$ with trace zero.