I hate to do this, but I cannot seem to remember/find a particular result that I thought was true. Forgive me if I have some points wrong, since this is the point of my asking.
I thought I remembered that the direct sum of an algebra and its opposite algebra was the universal enveloping algebra.
For a Lie algebra, the opposite Lie algebra is just that with negative bracket. But I don't see why the sum of these should be the universal enveloping algebra of the Lie algebra.
Edit: Qiaochu makes a great point below on the dimension in his comment.
I also don't think this should be true of some associative algebras, i.e. consider an associative algebra and its opposite(assuming the algebra is noncommutative, the opposite algebra is that with reversed multiplication, i.e. $a*b:=ba$), then considering the direct sum of these, it shouldn't be isomorphic to the enveloping algebra of the underlying Lie algebra of $A$, which is isomorphic to $A$.
I thought that I read the result in Dixmier, but I can't seem to find it. :/