sl2 triple ($[h,e]=2e,[h,f]=-2f,[e,f]=h$)

Over a field of characteristic zero, an sl2 triple consists of elements of a Lie algebra satisfying the displayed brackets. It defines a homomorphism from ${\mathfrak{sl}}_2$ by sending its standard generators to $h,e,f$. A nonzero homomorphism is injective because ${\mathfrak{sl}}_2$ is a simple Lie algebra. Root vectors and their coroot give such triples in a semisimple Lie algebra, and their representations supply finite weight strings.

Over a field of characteristic zero, in a finite-dimensional Lie algebra representation of an sl2 triple, start with a nonzero vector $X_0$ satisfying $f{X}_0=0$ and $h{X}_0=-\lambda X_0$, and set $X_n=e^n{X}_0$. The commutator identities give the displayed weight and lowering formulas by induction. If $X_N\ne 0$ and $X_{N+1}=0$, then $f{X}_{N+1}=(N+1)(\lambda -N)X_N$ 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.