# Week 10

## Session 1

### Pre-class exercises

Exercise:

True or false: If we use a sub-maximal torus, is it still true that $\mathfrak{h}=\mathfrak{g}_{0}$ ? If so, why? If not, what do we get instead?

## Session 2

### Pre-class exercises

Exercise:

Just by looking at the root diagrams, can you figure out the dimensions of the Lie algebras $\mathfrak{so}(5)$ and $\mathfrak{g}_{2}$ ?

Exercise:

Sketch the (3-dimensional) root system corresponding to your choice of Dynkin diagram from amongst: $A_{3}$ , $B_{3}$ , $C_{3}$ and $D_{3}$ .