It is well-known that for a complex semisimple Lie algebra $\mathfrak{g}$ with Cartan subalgebra $\mathfrak{h}$ and root system $\Phi$, there is a root space decomposition $\mathfrak{g}=\mathfrak{h}\oplus\bigoplus_{\alpha\in \Phi}\mathfrak{g}_\alpha$, where $\mathfrak{g}_\alpha:=\{x\in \mathfrak{g} : h\cdot x=\alpha(h)x, \forall h\in\mathfrak{h}\}$.
Meanwhile, $\Phi$ is a finite set of vectors and $\mathfrak{g}_\alpha$ is 1-dimensional.
My question: Does this implies any complex semisimple Lie algebra is finite dimensional? Or do I miss something?
You are missing the fact that the theorem about the existence of a root space decomposition is stated from the start as a theorem about finite-dimensional semisimple Lie algebras.