Relabelling indices in multiple summations

Suppose I have a summation that looks like
\begin{align}
\sum_{a=0}^{n/2} \sum_{b=0}^a \sum_{c=0}^{n-2b}\sum_{d=0}^c \alpha(a,b,c,d)f(c-2d),
\end{align}
where $\alpha$ and $f$ are functions of the indices $a,b,c,d$, and I want to change these summations in a particular way. I want the argument of the function $f$ to depend on just one index, and for the summation of that corresponding index to be independent of other indices. That is, I want to rewrite the above summation as
\begin{align}
\sum_w \sum_x \sum_y \sum_{z=-n/2}^{n/2} \alpha(w,x,y,y/2 - z) f(z),
\end{align}
where $w,x,y$ have replaced $a,b,c$. Furthermore, we see that $d = y/2 - z$.

I am having trouble figuring out the range of the three summations. I know there is some freedom involved here and I believe that the $w$ summation can be taken to be
\begin{align}
\sum_{w=0}^{n/2}
\end{align}
through its identification with the $a$-summation.

How can I systematically figure out what the remaining limits of summation are (for the $x$ and $y$ summations)?