Regular element of a semisimple Lie algebra

An element $x$ of a semisimple Lie algebra of rank $\ell$ is regular when its centralizer has the minimum possible dimension $\ell$.

For $t$ in a Cartan subalgebra $\mathfrak{t}$, the root-space decomposition givesThus $t$ is regular exactly when $\alpha (t)\ne 0$ for every root $\alpha$.