I sometimes see that people write group names in capital letters and sometimes in lowercase letters. Like the 3 dimensional special orthogonal group is sometimes written as $SO(3)$ and sometimes as $so(3)$. Is there any difference?
Usually the Lie groups are denoted by capital letters and the Lie algebra, which contains the generators of the Lie group, by lowercase letters. For instance, the Lie group SO(3) is associated with the Lie algebra so(3), which contains, for example in the case of the spin-1/2 representation, the Pauli matrices.