While I am reading a paper, I come across a difficulty. Here, we have a Lie group and we know its Lie algebra defined as
$[G_a,G_b]=f_{ab}^{\phantom{ab}c}G_c$
with $G_a\in\mathfrak g$. Under the adjoint representation, any element of the group, called $u$ can be expressed as
$u_a^{\phantom{a} b}=[e^{\lambda^c\rho(G_c)}]_a^{\phantom{a}b}=\delta_a^b-\lambda^cf_{ca}^{\phantom{ac} b}+\frac{1}{2!}\lambda^cf_{ca}^{\phantom{ac} e}\lambda^df_{de}^{\phantom{ac} b}-\cdots$
where the representation $[\rho(G_c)]_a^{\phantom{a}b}=-f_{ca}^{\phantom{ac} b}$ is used. The the author claims that the Maurer-Cartan form is given by
$\theta=\theta^aG_a$
with
$\theta^a=-\frac{1}{2}(\gamma^{-1})^{ab}\rho(G_b)_c^{\phantom{a} d}(u^{-1})_d^{\phantom{a} e}\mathrm du_e^{\phantom{a} c}$, and $\gamma_{ab}$ is the Cartan-Killing metric.
But how to show Maurer-Cartan form takes this complicated form? Can anybody give a hint? Thank you very much!