  We prove a theorem of splitting for the nonabelian tensor product $L \otimes
N$ of a pair $(L,N)$ of Lie algebras $L$ and $N$ in terms of its diagonal ideal
$L \square N$ and of the nonabelian exterior product $L \wedge N$. 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 $L \square N$
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 $\square$ 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.
