 sl2 triple

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.