Jacobi identity interpretations

The Jacobi identity in a Lie algebra $\mathfrak{g}$ has various interpretations that are more transparent, whence easier to remember, than the usual form$$ [x,[y,z]]+[y,[z,x]]+[z,[x,y]]=0.$$ One is the fact that the adjoint representation ¹$\ad:\mathfrak{g} \rightarrow \mathfrak{gl}(\mathfrak{g})$ really is a representation. Yet another way to formulate the identity is$$ \ad(x)[y,z]=[\ad(x)y,z]+[y,\ad(x)z],$$ i.e., $\ad(x)$ is a derivation on $\mathfrak{g}$ for all $x \in \mathfrak{g}$ .

Footnotes

... representation¹

Here, ``$\mathfrak{gl}(\mathfrak{g})$ '' means the space o endomorphisms of $\mathfrak{g}$ , viewed as a vector space, with Lie bracket on $\mathfrak{gl}(\mathfrak{g})$ being commutator.