Here is a proof I came up with any bilinear transformation with two fixed points say will be of the form when is neither unimodular nor the transformation is neither elliptical nor hyperbolic in the respective cases, hence loxodromic.
Now for all we know that and
we can express the transformation can be expressed as where and
where are elliptic and hyperbolic transformations respectively.
The notation used for may be confusing but what I mean is its just a matrix multiplication.
Let me know if this proof is ok.