= sl2 triple
{title2=$[h,e]=2e,\quad[h,f]=-2f,\quad[e,f]=h$}
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>.
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