Left and right fundamental differential form on Lie group

Suppose $G$ is a Lie group. $R_a$ and $L_a$ are right and left translations. Assume $e_i$ is a base of $T_eG$, the tangent space at the unit element $e$. $X_i$ denote the right-invariant vector field obtained by the right translation and $\tilde{X}_i$ by left translation. Let $w^i$ and $\tilde{w}^i$ denote the dual of $X_i$ and $\tilde{X}_i$. Then we have the Maurer-Cartan equation

$\displaystyle \begin{cases}dw^i=-\frac{1}{2}\sum\limits_{l,k=1}^n c^i_{lk}w^j\wedge w^k\\ c^i_{lk}+c^i_{kl}=0\end{cases}$

Here $c^i_{lk}$ are constants. Also we have the structure equation with respect to $\tilde{w}^i$. We see that there is a relation between $\tilde{c}^i_{lk}$ and $c^i_{lk}$, that is

$\tilde{c}^i_{lk}=-{c}^i_{lk}$

$\textbf{Proof: }$ Choose a local coordinate system $(U,x^i)$ at $e$. Then

$\displaystyle dy^i=\frac{\partial \phi^i(x,y)}{\partial x^j}\bigg|_{x=e}w^j$

where $\phi(x,y)=x\cdot y$. Apply exterior differentiation on both sides

$\displaystyle 0=\frac{\partial^2 \phi^i(x,y)}{\partial x^j\partial y^p}\bigg|_{x=e}dy^p\wedge w^j+\frac{\partial \phi^i(x,y)}{\partial x^j}\bigg|_{x=e}dw^j$

Letting $y=e$, we get

$\displaystyle dw^i=\frac{\partial^2 \phi^i(x,y)}{\partial x^k\partial y^l}\bigg|_{(e,e)}w^l\wedge w^k=-\frac{1}{2} \left(\frac{\partial^2 \phi^i(x,y)}{\partial x^l\partial y^k}\bigg|_{(e,e)}-\frac{\partial^2 \phi^i(x,y)}{\partial x^k\partial y^l}\bigg|_{(e,e)}\right)w^l\wedge w^k$

Then it is easy to see $\tilde{c}^i_{lk}=-{c}^i_{lk}$.

$\textbf{Remark: }$ see Chern’s lectures on differential geometry. p182