Let $L$ be a (differential) graded Lie algebra over a field $k$ of characteristic 0, and let $UL$ be the universal enveloping algebra of $L$.

The inclusion $L\hookrightarrow UL$ induces a morphism of the Lie algebra cohomology $H^*_{Lie}(L,L)\to H^*_{Lie}(L,UL)$. Can one deduce any properties of this map (like injectivity, surjectivity, etc...)?