How can one determine the dimension of a Lie algebra from the attached root system?

Question

Suppose we know the root system/Cartan matrix of a semisimple Lie algebra. Is there a formula that determines the dimension of the Lie algebra?

Thanks in advance.

Answer 1

I assume you are talking about a split semisimple Lie algebra. This is always the case if you work over an algebraically closed field of characteristic $0$, like $\Bbb C$.

Then it's part of the standard structure theory that such a Lie algebra $L$ decomposes as

$$ L \simeq H \oplus \bigoplus_{\alpha \in R} L_\alpha$$

where $H$ is a Cartan subalgebra, $R$ is the corresponding root system, and $L_\alpha$ is the root space to the root $\alpha$. It's also a standard fact that in this case, all $L_\alpha$ are one-dimensional, whereas the dimension of $H$ equals the rank of $R$. Which implies

$$dim(L) = rank(R) + \lvert R \rvert.$$