Isomorphism between Lie algebras

Question

This is an exercise (cf. exe 2.11 in the Erdmann and Wildon's book)

Define $$ gl_S(n, F) = \{ x\in gl(n,F)|\ x^t S = -Sx \} $$ Here $t$ is transpose.

Then let $T=P^tSP$ and show that $$ gl_T(n,F)=gl_S(n,F) $$ wher $P$ is invertible.

Thank you.

Answer 1

Let $x\in\mathfrak{gl}_{T}(n,\mathbb{F})$, then $x^tT=-Tx$. As $T=P^tSP$, so we get\[x^tP^tSP=-P^tSPx\iff (PxP^{-1})^tS=-SPxP^{-1}.\]Consequently, we can define $\phi:\mathfrak{gl}_{T}(n,\mathbb{F})\rightarrow \mathfrak{gl}_{S}(n,\mathbb{F})$ by $\phi(x)=PxP^{-1}$. The remained details are plain.