  For every finite $p$-group $G$ of order $p^n$ with derived subgroup of order
$p^m$, Rocco in \cite{roc} proved that the order of tensor square of $G$ is at
most $p^{n(n-m)}$. This upper bound has been improved recently by author in
\cite{ni}. The aim of the present paper is to obtain a similar result for a
non-abelian nilpotent Lie algebra of finite dimension. More precisely, for any
given $n$-dimensional non-abelian nilpotent Lie algebra $L$ with derived
subalgebra of dimension $m$ we have $\mathrm{dim} (L\otimes L)\leq
(n-m)(n-1)+2$. Furthermore for $m=1$, the explicit structure of $L$ is given
when the equality holds.
