Suppose $G$ be a Lie group over field $K$, and $\mathfrak{g}$ be its Lie algbera (again, over $K$).
Then, my question is in which case, they share(I'll explain this later) the same multiplication structure.
For instance, if $G$ is a matrix group over $K$, then every thing is done, since $G$ and $\mathfrak{g}$ shares the same multiplication structure in $\mathrm{Mat}_n(K)$ for some $n$. Of course, the results are not lying on neither $G$ nor $\mathfrak{g}$. But, the subalgebra generated by $G$ and $\mathfrak{g}$.
Are there good concepts about dealing with this? As we've seen, the matrix group is one of the important example. However, are there some non-trivial examples? What are the necessary, and sufficient conditions?
More precisely, are there some (possibly non-commutative and not-matrix) algebra $A$ over $K$ in (smooth category) and there are injective smooth $K$-algebra morphism from $K[G]$ to $A$ and injective $K$-Lie algebra morphism from $\mathfrak{g}$ to $A$, whose structures are also compatible with well-know results in smooth category, for example, $$ [X, Y] = \left. \frac{\partial}{\partial s}\right|_{s=0} \left.\frac{\partial}{\partial t}\right|_{t=0} \exp(tX) \exp(s Y) \exp(-tX) \exp(-sY) $$ in terms of $A$?