In differential geometry and Lie theory, generators are elements of the tangent space at the identity—the Lie algebra—which map into the Lie group via the exponential map rather than belonging to the group itself.
The claim states that generators belong strictly to the Lie algebra rather than the Lie group itself. Standard mathematical definitions and multiple papers in the literature (e.g., [0], [1], [2], [7]) confirm that generators form the Lie algebra (the tangent space at the identity), which generates the Lie group through the exponential map. Therefore, the claim is well-supported by standard mathematical principles and the provided literature.