Semisimple algebraic subgroup of $GL(V)$.

Question

Let $H$ be a semisimple algebraic subgroup of $GL(V)$ without compact factors (I am not sure if this part is relevant) where $V$ is a finite dimensional vector space. From a paper I have read, it follows that $H$ is contained in $SL(V)$. Is this true (or $H$ can be embedded in some way into $SL(V)$) and why?

Answer 1

I assume that $H$ is connected (and to simplify that $H$ is defined over $\mathbb{R}$) $[Lie(H),Lie(H)]=Lie(H)$, where $Lie(H)$ is the Lie algebra of $H$ since $[Lie(H),Lie(H)]\subset sl(V)$, $H\subset SL(V)$.

Answer 2

Let $H$ be a semisimple algebraic subgroup of $GL_n.$ Since $H$ is semisimple, $H$ is its own derived group, so it is in the derived group of $GL_n,$ which is $SL_n.$