= Construction of a principal sl2 triple
{title2=$e=\sum_i e_i,\quad h=2\rho^\vee,\quad f=\sum_i c_i f_i$}
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.
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