The structure of a solvable Lie groups admitting an Einstein left-invariant metric is, in a sense, completely determined by the nilradical of its Lie algebra. We give an easy-to-check necessary and sufficient condition for a nilpotent algebra to be an Einstein nilradical whose Einstein derivation has simple eigenvalues. As an application, we classify filiform Einstein nilradicals (modulo known ...

In the present paper we study generalized left derivations on Lie ideals of rings with involution. Some of our results extend other ones proven previously just for the action of generalized left derivations on the whole ring. Furthermore, we prove that every generalized Jordan left derivation on a 2-torsion free ∗-prime ring with involution is a generalized left derivation.

We study quadratic Lie algebras over a field K of null characteristic which admit, at the same time, a symplectic structure. We see that if K is algebraically closed every such Lie algebra may be constructed as the T∗-extension of a nilpotent algebra admitting an invertible derivation and also as the double extension of another quadratic symplectic Lie algebra by the one-dimensional Lie algebra...