I am following these lecture notes- and the Theorem 1.16 on page 12 reads
Suppose that $G ⊂ GL(n, R)$ is a subgroup and an embedded submanifold of $GL(n, R)$, and smooth structure on G is defined by the log-charts. Then the map $ξ ∈ \mathbf{g} → X_ξ ∈ l(G)$ is an isomorphism of the Lie algebras, where the Lie bracket on g is the Lie bracket of matrices (as in Theorem 1.7)
Note that $l(G)$ here mans the set of left invariant vector fields.
I don't get what the isomorphism is between. The only groups mentioned are $GL(n,R)$ and the subgroup $G$. But in general, $G$ can have a lower dimension than $GL(n,R)$ e.g. take U(2) of dimension 4, SU(3) of dimension 3. Then the corresponding lie algebras have dimensions 4 and 3, and so canto be isomorphic. So whih are the isomorphic lie algebras?
Some notation:
$\xi \in \mathbf{g}$ i.e. is an element of the lie algebra (though I am not sure if it here refers to the lie algebra of G or GL(n,R).
$X_\xi$ is defined at the vector field coorresponding to taking:
The left invariant diffeomorphism $L_g:h\in G \rightarrow gh \in G$
Which has a corresponding differential $(dL_g)_e : T_eG \rightarrow T_gG$
Then $X_\xi : h\in G \rightarrow (dL_h)_e \xi \in T_h G$
It is an isomorphism from $\mathfrak g$ (the Lie algebra of $G$) onto the Lie algebra $l(G)$ of all left-invariant vector fields in $G$.