Pre-Lie algebra

Remarks

In terms of the associator $(x, y, z) = (x \triangleright y) \triangleright z - x \triangleright (y \triangleright z)$, the condition says $(x, y, z) = (y, x, z)$. A right pre-Lie algebra asks instead for symmetry in the last two arguments. In a pre-Lie algebra, the commutator $[x, y] = x \triangleright y - y \triangleright x$ is a Lie bracket, so a pre-Lie algebra is Lie-admissible.

Example: $A = k[t]$ with $a \triangleright b = a b'$, where $'$ is the derivative. Then $(a \triangleright b) \triangleright c - a \triangleright (b \triangleright c) = a b' c' - a (b' c' + b c'') = -a b c''$, which is symmetric in $a$ and $b$. Every associative algebra is pre-Lie, since its associator is zero.

Non-example: a Lie algebra $\mathfrak{g}$ with $x \triangleright y = [x, y]$. Using the Jacobi identity, the difference of the two sides of the condition equals $[[x, y], z]$. For $\mathfrak{g} = \mathfrak{sl}_2$ with $[e, f] = h$ and $[h, e] = 2e$, take $x = e$, $y = f$, $z = e$: the difference is $[h, e] = 2e \neq 0$ when the characteristic of $k$ is not $2$.