Context-free grammar for"not-at-all" palindromes I need to bulid a context-free grammar for 

$\qquad \mathscr{L_4}=\{w\in\{a,b,c\}^* \mid w\text{ is not palindrome at all}\}$ 

Not palindrom at all:
We will say that a word $w$ is not palindrome at all if for all $i$ such that $1\leq i\leq |w|$ the $i$ letter from the beginning of $w$ is not the same like the $i$ letter of $w$ from the end, for example $abbacb\in \mathscr{L_4}$ but $ca\color{gray}bba\color{gray}bcb\notin\mathscr{L_4}$, the word must contain a even number of digits, because if there were an odd number of digits, so the middle digit "equals to itself". 

My attempt:
 Let's denote $S_{\text{pal}}$ from palindrome grammar and  $S_{\neg\text{pal}}$ for the not palindrome at all grammar 

$$\\S_{\neg\text{pal}}\to a S_{\neg\text{pal}}b\big|bS_{\neg\text{pal}}a\big|aS_{\neg\text{pal}}c\big|cS_{\neg\text{pal}}a\big|cS_{\neg\text{pal}}b\big|cS_{\neg\text{pal}}b\big|\epsilon$$

Similar question here.
I'm not sure if my attempt is correct or not
Just use something like:

$
S \rightarrow a S b \mid a S c \mid b S a \mid b S c \mid c S a \mid c S b \mid a b \mid a c \mid b a \mid b c \mid c a \mid c b
$

This ensures the $i$-th symbol from the beginning is different from the $i$-th from the end.