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two-step nilpotent Lie algebra; base; minimal system of generators; related sets; $H$-minimal system of generators
A Lie algebra $\mathfrak {g}$ is called two step nilpotent if $\mathfrak {g}$ is not abelian and $[\mathfrak {g},\mathfrak {g}]$ lies in the center of $\mathfrak {g}$. 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 field of complex numbers. Based on the study of minimal systems of generators, we choose an appropriate basis and give a complete classification of two step nilpotent Lie algebras of dimension $8$.
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