Decomposition of Adjoint Group of Semisimple Lie Group

Question

I have trouble understanding the beginning sentence of section $3$ of a paper by C.Moore: https://www.jstor.org/stable/2373052, which I quoted below:

Suppose now that $G$ is a (connected) semi-simple Lie group with Lie algebra $\mathfrak{g}_{0}$. Let $G^{*}$ be the adjoint group of $G$. Then one knows that $G^{*}$ is the direct product of its simple factors, $G^{*} = \prod_{i=1}^{n} G^{*}_{i}$, and that $\mathfrak{g}_{0}$ is the direct sum of the corresponding simple ideals $\mathfrak{g}_{0} = \prod_{i=1}^{n} (\mathfrak{g}_{0})_{i}$

My question is

$(1)$ How can $G^{*}$ be factored that way? Is it a general fact that adjoint group of a semisimple Lie group is semisimple? I can see how this is done if assuming $G$ is simply-connected or has discrete center, but the author didn't assume these.

$(2)$ How are the above $G_{i}^{*}$'s and $(\mathfrak{g}_{0})_{i}$'s related?

$(3)$ What are some good references to read so that I can better understand the above conclusion?

Your comments and insight are highly appreciated!

Answer 1

Firstly, for a general Lie algebra $\mathfrak{g}$ the adjoint group is a Lie group of the Lie algebra $\mathrm{ad}(\mathfrak{g})$ of inner derivations on $\mathfrak{g}$ but if $\mathfrak{g}$ is semisimple then $\mathrm{ad}(\mathfrak{g})=\mathfrak{g}$. So then the adjoint group is a Lie group of $\mathfrak{g}$ and so must be semisimple.

A semisimple Lie group always has discrete centre and moreover an adjoint group of a semisimple group has trivial centre. In fact you can think of all the connected Lie groups of a fixed semisimple Lie algebra on a scale between the simply connected group and the adjoint group. Each group is a quotient of the simply connected one by a subgroup of the centre and the adjoint group is the quotient by the whole centre. Not all of these groups can be factored into simple Lie groups but it is always true for the the simply connected one and the adjoint one.

The simple factors $G_i^*$ are of course Lie groups of the simple ideals $(\mathfrak{g}_0)_i$ and indeed must be adjoint groups themselves.