A Lie algebra $𝔤$ is called two step nilpotent if $𝔤$ is not abelian and $[𝔤,𝔤]$ lies in the center of $𝔤$. Two step nilpotent Lie algebras are useful in the study of some geometric problems, such as commutative Riemannian manifolds, weakly symmetric Riemannian manifolds, homogeneous Einstein manifolds, etc. Moreover, the classification of two-step nilpotent Lie algebras has been an important problem in Lie theory. In this paper, we study two step nilpotent indecomposable Lie algebras of dimension $8$ over the...

Let $C$ be an associative commutative ring with 1. If $a\in C$, then $aC$ denotes the principal ideal generated by $a$. Let $l,m,n$ be nonzero elements of $C$ such that $mn\in lC$. The set of matrices $\left(\begin{array}{ccc}{a}_{11}& a_{12}{a}_{21}& -a_{11}\end{array}\right)$, where $a_{11}\in lC$, $a_{12}\in mC$, $a_{21}\in nC$, forms a Lie ring under Lie multiplication and matrix addition. The paper studies properties of these Lie rings.