To see the relation to Cayley’s work of 1857 as described in my pdf Mathemagical Forests (MF), let the generators be represented by the vectors (infinitesimal Lie generators) and where , the derivative, and the multiplication operation by . (Call it the canonical pre-Lie operation, CPLO.) Then both sides of the AR reduce to .
In MF, initially the op is assigned to each vertex of forests of “naturally grown” rooted trees with the trees of each forest having the same number of vertices. Then starting with the leaves and working down, the CPLO is precisely the action of the resulting operator at each vertex on of the immediate lower vertex (see also Lagrange a la Lah). For example, if three leaves are attached by edges, or branches, directly to a lower vertex, the operator generated at that lower vertex is , which then operates via the CPLO on of the next lower vertex, or node. The associated forests represent the action of powers of infinitesimal Lie generators, , i.e., Lie vectors, and encode the repeated product differentiation rule, or Newton-Leibniz product rule, through the “natural growth” of the forests.
The resulting action of each side of the AR can be represented by a rooted tree with three nodes, or vertices, with two leaves and one root. is assigned to one leaf, to the other, and to the root. The resulting operation gives .
(1) Arthur Cayley, On the theory of the analytical forms called trees, Phil. Mag. 13 (1857), 172-176.
(2) An N-Category Cafe posting by Baez on week 299 and pre-Lie algebras
(4) Butcher series: A story of rooted trees and numerical methods for evolution equations by McLachlan, Modin, Munthe-Kaas, and Verdier
(5) Combinatorial Hopf algebras by Loday and Ronco (Pg. 28 has a description of a pre-Lie product in terms of nonplanar unreduced trees, and pg. 18, a bijection between planar binary (PB) and planar unreduced trees (PUT) that might be useful in translating between arguments made using one formulation into the other. )