Explicit formula Yang-Mills Lagrangian
I'm reading <PERSON>'s book Mathematical Gauge Theory and here in page $415$ we have
The local field strenght is given by
$$ (F^A)^a_{\mu \nu}=\partial_\mu A^a_\nu - \partial_v A^a_\mu +f_{a b c} A^b_\mu A^c_\nu $$
with structural constant defined by
$$[T_a,T_b]=\sum_{c=1}^r f_{a b c}T_c\quad \text{where ${T_c}$ is the base of the Lie Algebra $\mathfrak g$} $$
the YM-Lagrangian is $$ \mathcal L_{YM}[A]=-\frac{1}{4}(F^A)^a_{\mu \nu}(F^A)_a^{\mu \nu}$$
$$ =-\frac{1}{4}(\partial_\mu A^a_\nu -\partial\nu A^a_\mu)(\partial^\mu A^\nu_a - \partial^\nu A^\mu_a)$$
$$\quad -\frac{1}{2}f_{a b c}(\partial_\mu A^a_\nu -\partial_\nu A^a_\mu)A^{b \mu} A^{c \nu}$$
$$-\frac{1}{4} f_{a b c} f_{a d e} A^b_\mu A^c_{\nu} A^{d \mu} A^{e \nu} $$
I tried to write down this by hand, but I don't know what to do in order to obtain those two last terms in this expression. I'd appreciate any help !
* remark: I find out this question but they use trace formula and I'd like to use only the given structure equation for the curvature. 818