Definition 3

Assume that $\mathfrak{g}$ is semi-simple.

(i) A nilpotent element $x$ of $\mathfrak{g}$ is said to be principal if the dimension of Ker ad $x$ is the rank of $\mathfrak{g}$.

(ii) A simple element $h$ of $\mathfrak{g}$ is said to be principal if $h$ is regular and the eigenvalues of ad $h$ in an algebraic closure of $k$ belong to $2\mathbf{Z}$.

(iii) An $\mathfrak{s}\mathfrak{l}_2$-triplet $(x, h, y)$ of $\mathfrak{g}$ is said to be principal if the length of $\mathfrak{g}$, considered as a module over $kx+kh+ky$, is equal to the rank of $\mathfrak{g}$.