Let $G$ a compact connected (matrix) Lie group and $\mathfrak{g}$ its Lie algebra. Let $\{E_i\}$ be a basis of the Lie algebra thought as a vector space. Any element of the Lie group $G$ can be written as $e^X=\exp(a^iE_i)$ for some $X \in \mathfrak{g}$. For any Lie group there exists a left action $\lambda : \mathfrak{g} \times G \xrightarrow{} G$ given by $\lambda(X,p)=\exp(X)p$. So it is possible to define a map $\lambda_* : \mathfrak{g} \xrightarrow{} \mathcal{H}(G)$ from the Lie algebra to the algebra of vector spaces over $G$. Such map is defined as: $$ \lambda_* (X)(p):=\frac{d}{dt} \lambda(tX,p) |_{t=0} $$ for any $p \in G$.
It is possible to show that, in the case of a matrix Lie group, such a map is the multiplication map i.e.
$$ \lambda_*(X)(p)=Xp $$ where $Xp$ is the matrix multiplication.
I'm wondering if there is some nice formula for $$ E_i e^X=\lambda_*(E_i)(e^X) $$ In the case of a matrix Lie group the exponential map is the matrix exponential. Moreover, because $G$ is compact the Lie group exponential coincide with the Riemannian exponential w.r. to the Levi-Civita connection of any bi-invariant metric of $G$.
I have tried to expand the exponential, but I can't see any possible simplification.
There is some way in which such element can be expressed in terms of commutators of the Lie algebra?
If the answer is no, is there some way in which it can be expressed in terms of the Levi-Civita connection?