How to determine the number of roots and the dimension of a Lie algebra using Cartan matrix

Question

Let $\mathfrak{g}$ be a Lie algebra with the Cartan matrix $$ C=\left(\begin{array}{cc} 2 & -2\\ -1 & 2 \end{array}\right) $$

Question:

1. How can the number of roots of $\mathfrak{g}$ be determined from $C$?

2. How can it be shown that the dimension of $\mathfrak{g}$ is 10?

Note: this is an exam prep question.

Answer 1

The number of roots can be found by using the Cartan matrix to find the type of the root system. In this case the type is $B_2$ (or $C_2$), so there are $8$ roots.

The Lie algebra is the direct sum of a Cartan subalgebra and the weight spaces. In this case the Cartan subalgebra is two dimensional and the 8 weight spaces are one dimensional, so the dimension is 10.