LAUSR

We classify a class of 2-step nilpotent Lie algebras related to the representations of the Clifford algebras in the following way. Let $$J:{{\mathrm{\mathrm {Cl}}}}({\mathbb {R}}^{r,s})\rightarrow {{\mathrm{End}}}(U)$$J:Cl(Rr,s)→End(U) be a representation of the Clifford algebra $${{\mathrm{\mathrm {Cl}}}}({\mathbb {R}}^{r,s})$$Cl(Rr,s) generated by the pseudo Euclidean vector space $${\mathbb {R}}^{r,s}$$Rr,s. Assume that the Clifford module U is endowed with a bilinear symmetric non-degenerate real form $$\langle \cdot \,,\cdot \rangle _U$$⟨·,·⟩U making the linear map $$J_z$$Jz skew symmetric for any $$z\in {\mathbb {R}}^{r,s}$$z∈Rr,s. The Lie algebras and the Clifford algebras are related by $$\langle J_zv,w\rangle _U=\langle z,[v,w]\rangle _{{\mathbb {R}}^{r,s}}$$⟨Jzv,w⟩U=⟨z,[v,w]⟩Rr,s, $$z\in {\mathbb {R}}^{r,s}$$z∈Rr,s, $$v,w\in U$$v,w∈U. We detect the isomorphic and non-isomorphic Lie algebras according to the dimension of U and the range of the non-negative integers r, s.