Some properties on the tensor square of Lie algebras

In the present paper we extend and improve the results of \cite{bl, br} for the tensor square of Lie algebras. More precisely, for any Lie algebra $L$ with $L/L^2$ of finite dimension, we prove $L\otimes L\cong L\square L\oplus L\wedge L$ and $Z^{\wedge}(L)\cap L^2=Z^{\otimes}(L)$. Moreover, we show that $L\wedge L$ is isomorphic to derived subalgebra of a cover of $L$, and finally we give a free presentation for it.