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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?

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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.

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