Homework on Lie algebras

Christian Blohmann, my new office mate, is teaching a course on Lie theory. Recently, he covered the commutator algebra. Remember, that if A is a Lie algebra, then the set of all linear combinations of elements of the form [a,b] for a and b in A forms a subalgebra.

The question is: Do I really need the "linear combinations of" or can I get away without it. Your task is to find an example of a Lie algebra, where some
[a1,b1]+[a2,b2] cannot be written as [a,b]. Can you characterise algebras where this happens?

BTW, this is an exercise for you, I think we managed to figure out the solution. I just post it as I like it because it is not a standard exercise of the form "Show that the following assertion is true using the standard methods explained in class", it rather requires more creativity to solve it.

PS: Jacques Distler is right, TeX in a blog would be nice!