Lie Algebra associated to a lie group

Question

Given an infinite dimension vector space, let $G=I+End^f(V)$ where $End^f(V)$ is the ideal of finite rank endomorphism, and $H=G_1\subset G$ of endomorphisme of determinant $1$. How to calculate the lie algebra of $Aut(V)$, $G$ and $H$ ?

Answer 1

I think we can generelise the finite dimentional case, so we get $\mathfrak{gl}(Auto(V))=End(V)$, $\mathfrak{gl}(H)=H_0$ (traceless endomorphism in $H$).