Decomposition of the nonabelian tensor product of Lie algebras via the diagonal ideal

We prove a theorem of splitting for the nonabelian tensor product $L \otimes N$ofapair$(L,N)$ofLiealgebras$L$and$N$intermsofitsdiagonalideal$L \square N$andofthenonabelianexteriorproduct$L \wedge N$. A similar circumstance was described two years ago by the second author in the special case $N=L$. The interest is due to the fact that the size of $L\mathrm{\square }N$ influences strongly the structure of $L\otimes N$. Another question, often related to the structure of $L\otimes N$, deals with the behaviour of the operator $\mathrm{\square }$ with respect to the formation of free products. We answer with another theorem of splitting even in this case, noting some connections with the homotopy theory.