  A metaphor of Jean-Louis Loday describes Lie, associative, and commutative
associative algebras as ``the three graces'' of the operad theory. In this
article, we study the three graces in the category of $\mathfrak{sl}_2$-modules
that are sums of copies of the trivial and the adjoint representation. That
category is not symmetric monoidal, and so one cannot apply the wealth of
results available for algebras over operads. Motivated by a recent conjecture
of the second author and Mathieu, we embark on the exploration of the extent to
which that category ``pretends'' to be symmetric monoidal. To that end, we
examine various homological properties of free associative algebras and free
associative commutative algebras, and study the Lie subalgebra generated by the
generators of the free associative algebra.
