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The preceding Dynkin diagram identifies the root system $D_3$ with $A_3$. The classification of finite-dimensional complex simple Lie algebras by connected Dynkin diagrams therefore givesThis is the Isomorphism between so6 and sl4. Both algebras have dimension $15$, consistently with the classification.

The Killing form of a complex simple Lie algebra is nondegenerate. Since $B$ is also nondegenerate, there is a unique endomorphism $T$ of $\mathfrak{g}$ satisfyingInvariance of both forms givesso $T$ intertwines the adjoint representation. That representation is irreducible because its invariant subspaces are ideals. The Schur lemma therefore gives $T=cI$. Since $\kappa$ is nondegenerate, $c\ne 0$, andThis is the uniqueness of an invariant bilinear form on a simple Lie algebra.

Let $W\subseteq \mathfrak{g}$ be invariant under the Adjoint representation of a Lie algebra. Then $[\mathfrak{g},W]\subseteq W$, so $W$ is an ideal. If $\mathfrak{g}$ is a simple Lie algebra, its only ideals are $0$ and $\mathfrak{g}$. Hence the adjoint representation is irreducible. Compact type is compatible with the anti-Hermitian realization used below, although simplicity alone proves this adjoint irreducibility of a simple Lie algebra.

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.