For a simple complex Lie algebra $\mathfrak{ g}$, fixing a principal $\mathfrak{ sl}_2$-triple and highest weight vectors induces a basis of $\mathfrak{ g}$ as vector space. For $\mathfrak{ sl}_n$, we describe how to compute the Lie bracket in this basis using transvectants. This generalizes a well-known rule for $\mathfrak{ sl}_2$ 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.
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