Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 102 / 3 / a

For $t\in \mathfrak{t}$, the root-space decomposition gives its centralizerEvery root space is one-dimensional, so this centralizer has the minimum possible dimension $\ell =\dim \mathfrak{t}$ exactly when no summand on the right occurs. By the Regular element criterion in a Cartan subalgebra,