Corollary

Let E be a finite dimensional $\mathfrak{s}\mathfrak{l}(2, k)$-module.

(i) The endomorphism $H_E$ is diagonalizable and its eigenvalues are rational integers.

(ii) For any $p\in \mathbf{Z}$, let $E_p$ be the eigenspace of $H_E$ corresponding to the eigenvalue $p$. Let $i$ be an integer $\geq 0$. The map $X_{-E}^i|E_p: E_p\rightarrow E_{p-2i}$ is injective for $i\leq p$, bijective for $i=p$, and surjective for $i\geq p$. The map $X_{+E}^i|E_{-p}: E_{-p}\rightarrow E_{-p+2i}$ is injective for $i\leq p$, bijective for $i=p$, and surjective for $i\geq p$.

(iii) The length of E is equal to dim Ker$X_{+E}$ and to dim Ker$X_{-E}$.

(iv) Let $E'$ (resp. $E''$) be the sum of the $E_p$ for $p$ even (resp. odd). Then $E'$ (resp. $E''$) is the sum of the simple submodules of E of odd (resp. even) dimension; and $E = E'\oplus E''$. The length of $E'$ is dim $E_0$, and that of $E''$ is dim $E_1$.

(v) Ker$X_{+E}\cap$ Im$X_{+E}\subset \sum_{p>0}E_p$ and Ker$X_{-E}\cap$ Im$X_{-E}\subset \sum_{p<0}E_p$.

If E is simple, E is generated by a primitive element (Lemma 4), and it suffices to apply Propositions 1 and 2. The general case follows since every finite dimensional $\mathfrak{s}\mathfrak{l}(2, k$)-module is semi-simple.