rt.representation theory - Are complex semisimple Lie groups matrix groups?

Actually, my question is a bit more specific: Does every complex semisimple Lie group $G$ admit a faithful finite-dimensional holomorphic representation? [As remarked by Brian Conrad, this is enough to prove that $G$ is a matrix group (at least when it's connected) because $G$ can be made into an (affine) algebraic group over $mathbb{C}$ in unique way which is compatible with its complex Lie group structure, and under which every finite-dimensional holomorphic representation is algebraic. Furthermore, one can show that the image of a faithful representation would then be closed.]

Of course the analogous question for real semisimple Lie groups has a negative answer -- "holomorphic" having been replaced by "continuous", "smooth" or "real analytic" -- with the canonical counterexample being a nontrivial cover of $mathrm{SL}(2,mathbb{R})$.

For a connected complex semisimple Lie group $G$ I believe the answer is "YES." The idea is to piggy back off a 'sufficiently large' representation of a compact real form $G_mathbb{R}$; here by "compact real form" I'm referring specifically to the definition which allows us to uniquely extend continuous finite-dimensional representations of $G_mathbb{R}$ to holomorphic representations of $G$. I know (e.g. from the proof of Theorem 27.1 in D. Bump's Lie Groups) that such a definition is possible if we require $G$ to be connected (and I'd like to know if it's possible in general).

The details of the argument for connected $G$ are as follows. Consider the adjoint representation $mathrm{Ad} colon G to mathrm{GL}(mathfrak{g})$. Since $G$ is semisimple, $mathrm{Ad}$ has discrete kernel $K$. Consider next the restriction of $mathrm{Ad}$ to $G_mathbb{R}$. Observe that the kernel of this map is also $K$, for otherwise its holomorphic extension is different from the adjoint representation of $G$. Thus $K$ is finite, being a discrete, closed subset of a compact space. So by the Peter-Weyl theorem, we can find a representation $pi_0$ of $G_mathbb{R}$ that is nonzero on $K$. Extend $pi_0$ to a holomorphic representation $pi$ of $G$ and put $rho = pi oplus mathrm{Ad}$. Notice that $rho$ is a holomorphic representation of $G$ with kernel $kerpi cap K = 0$, which is what we were after.

What can we say if $G$ is disconnected?