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Choose simple root vectors with $[e_i,f_i]=h_i={\alpha }_i^{\vee }$, and let ${\rho }^{\vee }$ be the half-sum of positive coroots. It satisfies ${\alpha }_i({\rho }^{\vee })=1$. Expand $h=2{\rho }^{\vee }={\sum }_i{c}_i{h}_i$. For $i\ne j$, $[e_i,f_j]=0$, since the difference of distinct simple roots is not a root of a root system. Thus $[e,f]=h$, while $[h,e]=2e$ and $[h,f]=-2f$. These identities construct an embedded sl2 Lie algebra in any nonzero complex semisimple Lie algebra. Its raising vector is the principal nilpotent element with a nonzero component in each simple-root space.

Choose a Cartan subalgebra, simple roots ${\alpha }_1,\dots ,{\alpha }_r$, and root vectors $e_i,f_i$ normalized by $[e_i,f_i]=h_i={\alpha }_i^{\vee }$. LetHere the coroots lie in the Cartan subalgebra, and ${\alpha }_i({\rho }^{\vee })=1$. This last identity is the Weyl vector identity applied to the dual root system. For distinct simple roots, ${\alpha }_i-{\alpha }_j$ is not a root, hence $[e_i,f_j]=0$. ConsequentlyThese are exactly the sl2 Lie algebra relations. For nonzero $\mathfrak{g}$ the resulting homomorphism from ${\mathfrak{sl}}_2$ is nonzero and therefore injective, since ${\mathfrak{sl}}_2$ is simple. Its image is a Principal sl2 subalgebra. Equivalently its raising element is a principal nilpotent element, a nilpotent element with Lie algebra centralizer of dimension equal to the rank. The construction works for a semisimple Lie algebra with several simple factors by taking the triple in every factor, giving one diagonally embedded ${\mathfrak{sl}}_2$. This is the construction of a principal sl2 triple.

For a representation $V$, restrict to this subalgebra and take its formal character. If $V_{\mu }$ is the weight space for the original Cartan subalgebra, the resulting q-character of a highest-weight representation, or the analogous q-character for any finite-dimensional representation, isIt is independent of conjugating the chosen principal triple. In particular a root $\beta ={\sum }_i{k}_i{\alpha }_i$ has grading $\beta (h)=2{\sum }_i{k}_i$. Some conventions use $q^{h/2}$ instead; that convention divides all exponents here by two. The normalization above agrees with the ordinary sl2 Lie algebra character in the preceding part.

Put $h_0=\psi (h)$ and $e_0=\psi (e)={\sum }_{i=1}^{\ell }{e}_i$, where $0\ne e_i\in {\mathfrak{g}}_{{\alpha }_i}$. Since $[h_0,e_0]=2e_0$ and the root spaces are a direct sum, ${\alpha }_i(h_0)=2$ for every simple root. Every root has simple-root coefficients of one sign, so no root vanishes on $h_0$. The Regular element criterion in a Cartan subalgebra therefore shows that $h_0$ is regular.

Restrict the Adjoint representation of $\mathfrak{g}$ along $\psi$. By complete reducibility of semisimple Lie algebra representations, it is a direct sum of finite-dimensional sl2 Lie algebra modules. The $h_0$-eigenvalues on a root space are twice the heights of the roots, so they are all even. Each irreducible summand consequently has even highest weight, contains exactly one zero-weight vector, and has a one-dimensional kernel for the raising operator $e_0$ by the classification of finite-dimensional sl2 representations.

Because $h_0$ is regular, its zero-weight space in $\mathfrak{g}$ is precisely $\mathfrak{t}$ and has dimension $\ell$. There are therefore exactly $\ell$ irreducible summands, whenceThus $e_0$ is regular; equivalently it is a principal nilpotent element in the given Principal sl2 subalgebra.