Take for example $GL(2,\mathbb R)$, the group of $2\times2$ invertible matrices with real entries. By considering small variations from the identity, it is clear that one needs four parameters to parametrize this group, and hence we will need four "infinitesimal" generators. If I am thinking about this correctly, we could take as generators the matrices
$$ J_1 = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}, \quad J_2 = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}, \quad J_3 = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}, \quad J_4 = \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}. \tag{1}$$
But we could equally well take some inspiration from the Pauli matrices and use
$$ L_1 = \frac{1}{2} \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \quad L_2 = \frac{1}{2} \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}, \quad L_3 = \frac{1}{2} \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}, \quad L_4 = \frac{1}{2} \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}. \tag{2}$$
Both of these sets are linearly independent, so they seem like they should be fine, but they produce different Lie bracket (commutation) relations. Specifically, for the first set we have the (somewhat messy) relations
$$[J_1, J_2] = J_2, \quad [J_1, J_3] = -J_3, \quad [J_1, J_4] = 0, \quad [J_2, J_3] = J_1 - J_4, \quad [J_2, J_4] = J_2 \quad [J_3, J_4] = -J_3,\tag{3}$$ while for the second set we have $$[L_1, L_2] = L_3, \quad [L_1, L_3] = L_2, \quad [L_2, L_3] = L_1, \quad [L_i, L_4] = 0.\tag{4}$$
My question
When speaking about certain Lie algebras, I often read/hear people refer to the Lie bracket structure, say $[S_\alpha, S_\beta] = i\hbar\epsilon_{\alpha\beta\gamma}S_\gamma$, as the Lie algebra of the group (representation) in question. But, if my argument above is correct, it seems that this structure is not unique, but indeed depends on the generators chosen. So I can see two possible resolutions:
One (or both) of my suggested sets of generators is wrong. If this is the case, could you tell me why?
The Lie bracket structure for a given group is not unique, and those that say so are being sloppy with language somehow. Some of the more mathematically oriented sources seem to imply that the Lie algebra is actually a sort of tangent space about the identity transformation, with the generators as basis vectors. Then maybe it is this space that is unique, while the bracket structure is basis dependent?