The properties of an abstract Lie algebra are exactly those definitive of infinitesimal transformations, just as the axioms of group theory embody symmetry.
For example, in the case of infinitesimal rotations, the Lie algebra structure is that provided by the cross product, once a skew-symmetric matrix has been identified with a 3-vector.
This amounts to choosing an axis vector for the rotations; the defining Jacobi identity is a well-known property of cross products.
This then becomes a necessary condition on a smooth function F to have the homogeneity property; it is also sufficient (by using Schwartz distributions one can reduce the mathematical analysis considerations here).
Concentrating on the operator part, it shows that D is an infinitesimal transformation, generating translations of the real line via the exponential.