Can we prove that there are countably many isomorphism classes of compact Lie groups without appealing to the classification of simple Lie algebras?

It is a nontrivial fact that there are only countably many isomorphism classes of compact Lie groups. One can prove this by a series of reductions: first to the connected case, then to the simply connected case, then by classifying simple Lie algebras. Of course, this proof actually gives a much stronger classification result.

If I only want to prove that there are countably many isomorphism classes of compact Lie groups, can I work without appealing to the classification of simple Lie algebras? I have some ideas involving Tannaka's theorem but I haven't worked out a proof yet.

Solution 1:

1. It suffices to show that for each $n$, there are only countably many non-isomorphic compact $n$-dimensional Lie algebras.

2. Let $M_n$ be the space of $n$-dimensional Lie algebras: This is a certain affine variety. According to https://mathoverflow.net/questions/47447/deformations-of-semisimple-lie-algebras (see Ben Webster's answer, which does not use any structure theory), for every compact Lie algebra $g\in M_n$, all nearby Lie algebras are isomorphic to $g$. Therefore, the subset $C_n\subset M_n$ consisting of compact Lie algebras is discrete and, hence, countable.