When do two- or three-fund separation theorems hold?

We show that when asset returns satisfy a location-scale property (possibly conditionally as e.g. for multivariate generalized hyperbolic distribution) and the investor has law-invariant increasing preferences, optimal investment portfolio always exhibits two-fund or three-fund separation. As consequence, we recover many of (and two-fund) separation theorems have been derived in literature under very specific assumptions on preferences distributions. These are thus merely special cases general characterization result portfolios provide.