Definition 1

Let V be a $\mathfrak{g}$-module and $v\in V$. Then $v$ is said to be a primitive element of V if $v\not= 0$ and $v$ satisfies the conditions of Lemma 1.

A primitive element belongs to one of the $V^{\lambda}$. For all $\lambda \in \mathfrak{h}^*, V_{\pi}^{\lambda}$ denotes the set of $v\in V^{\lambda}$ such that $\mathfrak{b}_+v\subset kv$. Thus, the primitive elements of weight $\lambda$ are the non-zero elements of $V_{\pi}^{\lambda}$.