Let $G\subset SL(n,\mathbb{R})$ be a semisimple Lie group. In Dave Witte Morris's Introduction to Arithmetic Groups, he defines $G$ to be algebraically simply connected if, for all $\ell\geq 1$ and every Lie algebra homomorphism $\beta: \mathfrak{g}\to \mathfrak{sl}(\ell,\mathbb{R})$, there exists a Lie group homomorphism $\alpha:G\to SL(\ell,\mathbb{R})$ such that $d_e\alpha = \beta$.
It's a standard theorem (see Lie group - Lie algebra correspondance) that any simply connected group is algebraically simply connected. However, $SL(n,\mathbb{R})$ is an example which shows the converse is false. Indeed, one can extend any lie algebra homomorphism $\mathfrak{sl}(\ell,\mathbb{R}) \to \mathfrak{sl}(\ell,\mathbb{R})$ to a homomorphism $\mathfrak{sl}(\ell,\mathbb{C}) \to \mathfrak{sl}(\ell,\mathbb{C})$ and then use that $SL(n, \mathbb{C})$ is simply connected.
Witte claims that any semisimple $G\subset SL(n, \mathbb{R})$ has a finite cover which is algebraically simply connected. Any ideas why this is true? I've tried reading standard the proofs for why simply connected implies algebraically simply connected, but haven't been able to modify them to show this. Any suggestions would be appreciated. Thanks!