Lie triple system

Remarks

Example: let $\mathfrak{g}$ be a Lie algebra with bracket $[\cdot, \cdot]$ and put $[x, y, z] = [\,[x, y], z]$. Clause 1 holds because $[x, x] = 0$. Clause 2 is the Jacobi identity. Clause 3 says that the map $z \mapsto [\,[u, v], z]$ acts as a derivation of the triple product, and this holds because $\mathrm{ad}_{[u, v]}$ is a derivation of the bracket. The same formulas hold on any subspace of $\mathfrak{g}$ closed under the triple product, for example on the $-1$ eigenspace of an involutive automorphism of $\mathfrak{g}$.

Non-example: on $\R^n$ with the dot product, the trilinear map $[x, y, z] = \ip{x}{y} z$ fails clause 1, since $[x, x, z] = \abs{x}^2 z \ne 0$ for nonzero $x$ and $z$.

Clause 1 implies the skew-symmetry $[x, y, z] = -[y, x, z]$, by expanding $[x + y, x + y, z] = 0$.