= Finite sl2 lowest-weight ladder
{title2=$hX_n=(2n-\lambda)X_n,\quad fX_n=n(\lambda-n+1)X_{n-1}$}
In a finite-dimensional representation of an <sl2 triple>, start with a nonzero vector $X_0$ satisfying $fX_0=0$ and $hX_0=-\lambda X_0$, and set $X_n=e^nX_0$. The <commutator> identities give the displayed weight and lowering formulas by induction. If $X_N\ne0$ and $X_{N+1}=0$, then $fX_{N+1}=(N+1)(\lambda-N)X_N$ forces $\lambda=N$, a nonnegative <integer>. The nonzero terminal vector is essential: choosing a later index after the ladder has already vanished gives no constraint. Applied to the <Adjoint representation> of a <semisimple Lie algebra>, this argument proves integrality of <Cartan integers>.
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