I am unsure if the posting of comments is brought to the attention of the Math Overflow community, so I am creating this post simply to generate interest
The bug in the example given in the Salvo's answer above . Please see my latest comment below the answerhas finally been fixed! It is now entirely correct.
My original concern is was that, due to the requirement that the Jacobi Identity be satisfied, an infinite-dimensional Lie algebra (or even a finite-dimensional one) cannot be defined simply by playing with relations among basis vectors . (which is what Salvo and I think that did initially, thus leading to a number of false starts). Hence, one needs to either work with "natural natural" examples " of infinite-dimensional Lie algebras (as Professor Humphreys has mentioned in his comment below the statement wording of my question) or to labor struggle really hard with the Jacobi Identity in order to generate non-natural "non-natural" examples.