Let $H$ be a Lie group and $\mathfrak{h}$ its Lie algebra. Given a smooth function $v: H \to \mathfrak{h}$, define the vector field $\bar{v} : H \to TH$, $h \mapsto d(R_{h})_{e} v(h)$, where $R_{h} : H \to H$ is right multiplication by $h$.
Given two smooth functions $v, w: H \to \mathfrak{h}$, I want to find $[\bar{v}, \bar{w}]|_{e}$ in terms of $[v(e),w(e)]_{\mathfrak{h}}$ (where $[ , ]_{\mathfrak{h}}$ is the bracket on $\mathfrak{h}$).
Here's what I've done: In some neighbourhood of the identity $e$, we can write $v(h) = v_{0} + h^{i}v_{i}(h)$ using Taylor's theorem (where $h^{i}$ are coordinate functions centered at $e$). Then $\bar{v}(h) = v_{0}^{R}|_{h} + h^{i} \tilde{v}_{i}(h)$, where $v_{0}^{R}|_{h} = d(R_{h})_{e}v_{0}$ is the right-invariant vector field, and $\tilde{v}_{i}(h) = d(R_{h})_{e}v_{i}(h)$. Similarly for $w$.
Then $[\bar{v}, \bar{w}]|_{0} = [v_{0}^{R},w_{0}^{R}]|_{0} + v_{0}^{i}w_{i}(0) - w_{0}^{i}v_{i}(0)$, where for example $v_{0}^{i}$ is the $i^{th}$ component of $v_{0}$ (i.e. $v_{0}(h^{i})$). This can be rewritten as follows:
$[\bar{v}, \bar{w}]|_{0} = [v_{0}^{R},w_{0}^{R}]|_{0} + v^{i}(0)\frac{\partial w}{\partial h^{i}}(0) - w^{i}(0) \frac{\partial v}{\partial h^{i}}(0)$.
The last part of this equation $v^{i}(0)\frac{\partial w}{\partial h^{i}}(0) - w^{i}(0) \frac{\partial v}{\partial h^{i}}(0)$ looks like a bracket $[v,w]|_{0}$, but $v$ and $w$ are not vector fields. I thought it would be helpful to rewrite $v$ as $\theta^{R} \bar{v}$, where $\theta^{R}$ is the right-invariant Maurer-Cartan form, but I havn't been able to use it.