The left-invariant vector fields on a Lie Group $G$ constitute a canonical Lie Algebra $\mathfrak{g}$ for $G$, Lie($G$).
With some adequate definition, one can also define a bracket on a vector space such that one obtains a Lie Algebra, e.g. commuting matrices for $\mathfrak{gl}(n)$, and in this case Lie($GL(n)$)$\simeq \mathfrak{gl}(n)$.
How can one find (describe) all possible isomorphic Lie Algebras to the canonical one, Lie($G$)?
The "variety" of all Lie algebra structures $\mathcal{L}_n(K)$ over a given vector space $V$ of dimension $n$ has points $(c_{ij}^k)\in K^{n^3}$, satisfying skew-symmetry and the Jacobi identity. If $(e_1,\ldots e_n)$ is a basis of $V$, then $[e_i,e_j]=\sum_{k=1}^n c_{ij}^ke_k$. So a fixed Lie algebra $L$ of dimension $n$ is given there by all sets of structure constants defining a Lie algebra isomorphic to $L$. The group $GL_n(K)$ acts on $\mathcal{L}_n(K)$ by base change. In this sense, one could say that "all possible isomorphic Lie algebras to the canonical one" are given by such sets of structure constants.