Some braces of cardinality p4 and related Hopf-Galois extensions

We describe all Fp-braces of cardinality p4 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 Fp-braces