Some braces of cardinality p(4) and related Hopf-Galois extensions

We describe all F-p-br aces of cardinality p(4) which are not right nilpotent. The constructed braces are solvable and prime, and contain a non-zero strongly nilpotent ideal. We use the constructed braces to construct examples of finitely dimensional pre-Lie algebras which are left nilpotent but not right nilpotent. We also explain some well known results about the correspondence between braces and Hopf-Galois extensions using the notion of Hopf-Galois extensions associated to a given brace. This can be applied to the constructed F-p-braces.