Structure constants for simple Lie algebras from principal ${\mathfrak{s}\mathfrak{l}}_2$-triple

For a simple complex Lie algebra $\mathfrak{g}$, fixing a principal ${\mathfrak{s}\mathfrak{l}}_2$-triple and highest weight vectors induces a basis of $\mathfrak{g}$ as vector space. For ${\mathfrak{s}\mathfrak{l}}_n$, we describe how to compute the Lie bracket in this basis using transvectants. This generalizes a well-known rule for ${\mathfrak{s}\mathfrak{l}}_2$ using Poisson brackets and degree 2 monomials in two variables. Our proof method uses a graphical calculus for classical invariant theory. Other Lie algebra types are discussed.