About root space decomposition of complex semisimple Lie algebra

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?

• However, I read two books of Humphreys: Introduction to Lie Algebras and Representation Theory and Representations of Semisimple Lie Algebras in the BGG Category O. The sections about semisimple Lie algebra $\mathfrak{g}$ has not assume $\mathfrak{g}$ is finite dimensional but just assume $\mathfrak{g}$ is a complex semisimple Lie algebra with Cartan subalgebra $\mathfrak{h}$ . Results about root space decomposition are also included in these two books. Why is that? Do you have any references about the assumption "Being finite dimensional semisimple Lie algebra"? May 8 '19 at 10:32
• In Humphrey's textbook, in iw written, on page 1, “In this book we shall be concerned almost exclusively with Lie algebras $L$ whose underlying vector space is finite dimensional over $F$. This will always be assumed, unless otherwise stated. May 8 '19 at 10:34