# Abelian Lie group representation

I have the following doubt:

If I have an abelian Lie algebra with 3 generators (let's call them $$t_1, t_2, t_3$$), by definition of adjoint representation all of them will be equal to the $$3\times 3$$ zero-matrix and by exponentiation, the group elements will all be equal to the $$3\times 3$$ identity matrix.

Nevertheless, the generators are supposed to be different, so our algebra is three dimensional, but the adjoint representation made them all equal, so this looks like a lost of dimensions. Does this make sense or maybe the adjoint representation is not a good idea here?

There is nothing peculiar in this situation. What happens is that the adjoint representation is not a faithful representation. That's all. By definition of center, all elements of the center of a Lie algebra act is $$0$$ in its adjoint representation.