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 representation¹¹Here, “ $\\mathfrak{gl}(\\mathfrak{g})$ ” means the space oendomorphisms of $\\mathfrak{g}$ , viewed as a vector space, with Liebracket on $\\mathfrak{gl}(\\mathfrak{g})$ being commutator. $\\operatorname{ad}:\\mathfrak{g}\\rightarrow\\mathfrak{gl}(\\mathfrak{g})$ really is a representation. Yet another way to formulate the identity is

\\operatorname{ad}(x)[y,z]=[\\operatorname{ad}(x)y,z]+[y,\\operatorname{ad}(x)z],

i.e., $\\operatorname{ad}(x)$ is a derivation on $\\mathfrak{g}$ for all $x\\in\\mathfrak{g}$ .