What is the process of converting the function of four variables into a function with two variables

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When I have this function $$f(x_1,x_2,x_3,x_4)=x_1x_3+x_2x_4$$

I convert it to a function with two variables

As an example $x_1=x_2=x,x_3=x_4=y$ I get $f(x,y)=2xy$

But I have many cases

1- $x_1=x_3=x,x_2=x_4=y$ I get $f(x,y)=x^2+x^2$

2-$x_1=x_4=x,x_3=x_4=y$ I get $f(x,y)=2xy$

...

I need to calculate all functions resulting from the restriction

There will be a repetitive process of hypothesis for variables.

What is the process of converting the function of four variables into a function with two variables ?

Is there a mathematical basis for this process as a theory?

Thanks for the help