On pseudo-Euclidean Lie algebras whose Levi-Civita product is left Leibniz

We study a class of Lie algebras which contains the class of quadratic Lie algebras and the class of Milnor Lie algebras, namely, Lie algebras endowed a pseudo-Euclidean metric such its Levi-Civita product is left Leibniz. We call them Levi-Civita left Leibniz Lie algebras LCLL for short. We show that a Lie group (G,h) endowed with a left invariant pseudo-Riemannian such that the corresponding Lie algebra is LCLL is complete and locally symmetric. Moreover, there is a pointed Lie rack structure on G whose corresponding left Leibniz algebra is given by the Levi-Civita product. Euclidean LCLL Lie algebras are the product of a quadratic Lie algebra and a flat Euclidean Lie algebra. We develop an adapted version of the process of double extension to construct LCLL Lie algebras. We show that Lorentzian or flat LCLL Lie algebras can be obtained by this process.