  Eastwood and Ezhov generalized the Cayley surface to the Cayley hypersurface
in each dimension, proved some characteristic properties of the Cayley
hypersurface and conjectured that a homogeneous hypersurface in affine space
satisfying these properties must be the Cayley hypersurface. We will prove this
conjecture when the domain bounded by a graph of a function defined on $\R^n$
is also homogeneous giving a characterization of Cayley hypersurface. The idea
of the proof is to look at the problem of affine homogeneous hypersurfaces as
that of left symmetric algebras with a Hessian type inner product. This method
gives a new insight and powerful algebraic tools for the study of homogeneous
affine hypersurfaces.
