When discussing representation theory of Lie algebras $\rho : \mathfrak{g} \mapsto M_n(\mathbb{C})$, one is often interested in the product of two such elements $\rho(X)\rho(Y)$, meaning that one is also considering the 'multiplicative properties' of the Lie algebra representation. Of course, in general such matrices will be singular and so no group structure may be defined on the image of $\rho$.
Similarly, one could also consider 'additive' properties of a Lie group representation $\Phi : G \mapsto M_n(\mathbb{C})$, i.e. elements of the type $\Phi(g_1) + \Phi(g_2)$. What kind of structure could one get this way? What is the resulting subspace of $M_n(\mathbb{C})$? What is the Lie algebra obtained by taking linear combinations of nested commutators, $\Phi(g_1)\Phi(g_2) - \Phi(g_2)\Phi(g_1)$? How do these properties depend on the specific representation/type of Lie algebra (e.g. connectedness, semi-simplicity/solvability, highest weight, etc.)
The set should probably reduce just to the elements of the vector space, since $\Phi(g_1)\Phi(g_2) = \Phi(g_1g_2)$ by definition of a representation; but the Lie algebraic properties might be non-trivial.
Also, this Lie algebra $\text{Im}(\Phi(G))$ induced by the representation should also include the representation of the Lie algebra $\mathfrak{g} = \text{Lie}(G)$, since for elements $g\in G$ close to the identity one should simply have elements of $\phi(\mathfrak{g})$: $\Phi(g) \approx \mathbb{1}_n + \phi(x)$. However I'm not sure whether the two Lie algebras $\text{Im}(\Phi(G))$ and $\text{Lie}(G)$ coincide, or the latter is only a subalgebra of the former.
