I try to understand why by definition
$[c0,c1,...cn]=[c0,[c1,...cn]]$ and also
$[c0,c1,...cn]=[c0,c1,...c(n-2),[c(n-1),cn]]$ .
But not
$[c0,c1,...cn]=[c0,c1,[c2,...cn]]$ for example.
I presume that your "[c0, c1, ..., cn]" means the continued fraction $\displaystyle \frac{c0}{1+ \frac{c1}{1+ \cdot\cdot\cdot}+ cn"}}$. It should be clear that you can factor out that top "c0" to get $\displaystyle c0\frac{1}{1+ \frac{c1}{1+ \cdot\cdot\cdot+ cn}}$ which is your "[c0, [c1, ..., cn]]".
Try doing that with n= 2 or 3 to see what is happening.