Remark 1

The adjoint representation of $\mathfrak{s}\mathfrak{l}(2, k)$ defines on $\mathfrak{s}\mathfrak{l}(2, k)$ the structure of a simple $\mathfrak{s}\mathfrak{l}(2, k$)-module. This module is isomorphic to V(2) by an isomorphism that takes $u^2$ to $X_+, 2uv$ to $-H$, and $v^2$ to $X_-$.