Capability of nilpotent Lie algebras of small dimension

Given a nilpotent Lie algebra $L$ of dimension $\le 6$ on an arbitrary field of characteristic $\mathrm{\ne }2$, we show a direct method which allows us to detect the capability of $L$ via computations on the size of its nonabelian exterior square $L\wedge L$. For dimensions higher than $6$, we show a result of general nature, based on the evidences of the low dimensional case, focusing on generalized Heisenberg algebras. Indeed we detect the capability of $L \wedge L$viathesizeoftheSchurmultiplier$M(L/Z^\wedge(L))$of$L/Z^\wedge(L)$, where $Z^{\wedge }(L)$ denotes the exterior center of $L$.