A Witt algebra W is an infinitedimensional Liealgebra defined by the generator relations:
W: $[L_{j},L_{k}]:=(jk)\cdot L_{j+k}$
And my first thought was: What about the analogous algebra defined by
W': $[L_{j},L_{k}]:=j\cdot{L_k}k\cdot{L_j}$
(Yes, I checked, the Jacobi relation is fulfilled.)
There are about two possibilities: a) W' is uninteresting, question can be closed. b) W' is interesting and someone had the idea before me. Inventor's name suffices then. :) (Unfortunately, there is no way to google such a formula.)
Interesting/uninteresting is a very subjective thing, so let me try to just say several things that I see immediately.
0) This algebra, unlike the Witt algebra, does not have any [obvious] grading, and this makes it in a sense less interesting for some purposes.
1) This algebra, unlike the Witt algebra, comes from antisymmetrising an associative product $L_j\circ L_k=j L_k$ (in the case of the Witt algebra, it is not associative, but only satisfied the so called [left] preLie identity $(a\circ b)\circ ca\circ(b\circ c)=(b\circ a)\circ cb\circ(a\circ c)$. So in a sense it is more unusual: Lie algebras do not frequently arise that way.
2) The associative product above is not only associative but it does in fact satisfy the so called [left] permutative identity $a\circ b\circ c=b\circ a\circ c$ (so this algebra is very close to commutative; it is only noncommutative because there is no unit element). The operad of permutative algebras is the Koszul dual of the operad of preLie algebras, so it is kind of an amusing coincidence.
3) This Lie algebra contains a onedimensional ideal spanned by $L_0$, and the quotient by this ideal, after a linear change of basis $M_j=\frac{L_j}{j}$, has simpler commutation relations $[M_j,M_k]=M_kM_j$ ($j,k\ne 0$).

2$\begingroup$ Such are the (technical) things I simply can't know as an amateur when I pull a formula from my hat based more on aesthetic reasons than anything. Especially 3 I find very interesting, this should greatly simplify the search whether it appeared in literature before. $\endgroup$ Jul 9 '16 at 8:24