These algebras were considered by Ramaiengar Sridharan a long time ago. See [Sridharan, R. Filtered algebras and representations of Lie algebras. Trans. Amer. Math. Soc. 100 1961 530--550. [MR0130900](http://www.ams.org/mathscinet-getitem?mr=MR0130900) (24 #A754)]

If the map you are using to twist is not a Chevalley-Eilenberg cocycle, then things are ugly. In particular, you do not get a PBW-basis of the algebra (the cocycle condition is equivalent to the BPW property, in fact; this was in the general context of quadratic Koszul algebras a few years ago)

By the way: I call them *Sridharan enveloping algebras*, and I have heard others do the same.