Algebraic approach to Rump's results on relations between braces and pre-lie algebras

In 2014, Wolfgang Rump showed that there exists a correspondence between left nilpotent right Double-struck capital R-braces and pre-Lie algebras. This correspondence, established using a geometric approach related to flat affine manifolds and affine torsors, works locally. In this paper, we explain Rump's correspondence using only algebraic formulae. An algebraic interpretation of the correspondence works for fields of sufficiently large prime characteristic as well as for fields of characteristic zero.