Covariant differential on p forms On Peter Li's book  Geometric Analysis on page 19 (http://www.im.ufrj.br/andrew/GR14-2/Lecture%20Notes%20on%20Geometric%20Analysis.pdf) I can't understand the following line 
$\begin{array}{l}
da_{i_1\cdots i_p-1,j}\wedge w_j+a_{i_1\cdots i_p-1,k}w_{kj}\wedge w_j\\
=a_{i_1\cdots i_p-1,jk}w_k\wedge w_j-a_{i_1\cdots k_a\cdots i_p-1,j} w_{k_ai_a}\wedge w_j
\end{array}$
In fact my main doubt is about the second term on the RIGHT hand side, that I can't understand how it appears.
For more explanation  about the notation, the book is found on http://www.im.ufrj.br/andrew/GR14-2/Lecture%20Notes%20on%20Geometric%20Analysis.pdf, page 19. Thanks for help.
 A: In the formula at pag. 19 after "Exterior differentiating this, we have" the author used the Cartan structural equation
$$d\omega_k = \omega_{kj}\wedge \omega_j $$
which defines the $\omega_{kj}$'s, given the basis $\{\omega_k\}$ of the cotangent space $T^*_xM$, dual to the basis $\{e_k\}$ on $T_xM$, for all $x\in M$.
In particular
$$ d(a_{i_1\cdots i_{p-1},j}\wedge w_j)=da_{i_1\cdots i_{p-1},j}\wedge \omega_j+ a_{i_1\cdots i_{p-1},j}d \omega_j = da_{i_1\cdots i_{p-1},j}\wedge \omega_j + a_{i_1\cdots i_{p-1},k}d \omega_k = da_{i_1\cdots i_{p-1},j}\wedge \omega_j + a_{i_1\cdots i_{p-1},k} \omega_{kj}\wedge \omega_j $$
EDIT
After a change in the OP, I update my answer. We know that
$$d(a_{i_1\cdots i_{p-1},j}\wedge w_j) = da_{i_1\cdots i_{p-1},j}\wedge \omega_j + a_{i_1\cdots i_{p-1},k} \omega_{kj}\wedge \omega_j; $$
we begin by proving that
$$d(a_{i_1\cdots i_{p-1},j}\wedge w_j) =  da_{i_1\cdots i_{p-1}} \wedge \omega_{j_a i_a} + 
a_{i_1\cdots j_a\cdots i_{p-1}} \wedge \omega_{j_a k} \omega_{ki_a} + \\
\frac{1}{2} a_{i_1\cdots j_a\cdots i_{p-1}}\mathcal R_{j_a i_a k l} \omega_l \wedge \omega_k,  $$
i.e. the second equation at pag. 19 in the reference.
By definition of the covariant derivative (first definition at pag. 19), we have
$$ d(a_{i_1\cdots i_{p-1},j}\wedge \omega_j) = d^2 a_{i_1\cdots i_{p-1}} + 
da_{i_1\cdots j_a \cdots i_{p-1}} \omega_{j_a i_a} + 
a_{i_1\cdots j_a \cdots i_{p-1}}d \omega_{j_a i_a} = da_{i_1\cdots j_a \cdots i_{p-1}} \omega_{j_a i_a} + a_{i_1\cdots j_a \cdots i_{p-1}}d \omega_{j_a i_a}.~~(*) $$
All we need is to apply the Cartan second structural equation (pag. 16)
$$d \omega_{j_a i_a}= \omega_{j_a k}\wedge \omega_{k i_a} + \frac{1}{2}\mathcal R_{j_a i_a k l} \omega_l \wedge \omega_k $$
to the r.h.s. of (*) and arrive at the result. 
We can move to the original equation in the OP. The second term on the r.h.s. in such equation follows from the use of the definition of covariant derivative on $d a_{i_1\cdots i_{p-1},j}$.In  fact
$$ a_{i_1\cdots i_{p-1},jk}\omega_k =d a_{i_1\cdots i_{p-1},j} + 
 a_{i_1\cdots k_a\cdots i_{p-1},j}\omega_{k_a i_a}. $$ 
The above equation is the definition of the covariant derivative of the symbols 
$a_{i_1\cdots i_{p-1},j}$. Reversing it we arrive at
$$ d a_{i_1\cdots i_{p-1},j} = a_{i_1\cdots i_{p-1},jk}\omega_k -
 a_{i_1\cdots k_a\cdots i_{p-1},j}\omega_{k_a i_a}, $$
as desired.
