Coproduct of Lie algebras

Question

Fix a commutative ring $k$ and look at the category of Lie algebras over $k$. How do coproducts in that category look like? Notice that what is usually called the "direct sum" of Lie algebras is not the coproduct. The underlying set of the coproduct $L \sqcup L'$ of two Lie algebras $L,L'$ will contain (arbitrarily?) nested Lie brackets such as $[[a,b],c]$ with $a,c \in L$, $b \in L'$. I wonder if there is any concise description, i.e. a sort of normal form, similar to the one which is available for the coproduct of groups.