Can the bilinear form of an inner product be extended to both sides?

Question

The bilinear axiom is:

 <cu + dv,w> = c<u,w> + d<v,w>
 <u,cv + dw> = c<u,v> + d<u,w>

Where c and d are scalars and u, v, and w are vectors.

Can this be extended to something like

 <cu + dv, ew + fx> = ?

Answer 1

The same as above: $$<cu+dv,ew+fx>=c<u,ew+fx>+d<v,ew+fx>=c(e<u,w>+f<u,x>)+d(e<v,w>+f<v,x>)$$

Answer 2

Sure:$$\langle cu+dv,ew+fx\rangle=ce\langle u,w\rangle+cf\langle u,x\rangle+de\langle v,w\rangle+df\langle v,x\rangle.$$You get this applying what you call “bilinear axioms”.