Capability of Nilpotent Lie Algebras of Small Dimension

Given a nilpotent Lie algebra L of dimension $$\le 6$$ on an arbitrary field characteristic $$\ne 2$$ , we show direct method to detect whether is capable or not via computations the size its nonabelian exterior square $$L \wedge L$$ . For dimensions higher than 6, result general nature, based evidences low dimensional case, but also large families algebras, namely generalized Heisenberg algebras. Indeed, capability Schur multiplier $$M(L/Z^\wedge (L))$$ $$L/Z^\wedge (L)$$ where $$Z^\wedge denotes center L.