I would like to ask about the setting of the book **"Representations of Semisimple Lie Algebras in the BGG Category $\mathcal{O}$"** by Humphreys. I would like to know whether the semisimple Lie algebra $\mathfrak{g}$ is finite dimensional or not. I search the book and did not find any assumption about the semisimple Lie algebra $\mathfrak{g}$ being finite dimensional. However, the book mentions root space decomposition of $\mathfrak{g}$, which depends on the finite dimensionality of $\mathfrak{g}$ by Humphreys' other book **"Introduction to Lie Algebras and Representation Theory"**. Also a Cartan subalgebra $\mathfrak{h}$ of $\mathfrak{g}$ is mentioned to be finite dimensional on p.1 of **"Representations of Semisimple Lie Algebras in the BGG Category $\mathcal{O}$"**.

**My questions:**

- Is the semisimple Lie algebra $\mathfrak{g}$ finite dimensional?
- If $\mathfrak{g}$ is not assumed to be finite dimensional, do we still have the root space decomposition of complex semisimple Lie algebra with respect to a Cartan subalgebra?
- If $\mathfrak{g}$ is not assumed to be finite dimensional, does the root space decomposition imply that any complex semisimple Lie algebra with a Cartan subalgebra is finite dimensional?
- If $\mathfrak{g}$ is not assumed to be finite dimensional, how do we prove $\mathfrak{h}$ is finite dimensional?