The affine homogeneous hypersurface in ${mathbb{R}}^{n+1}$, which is a graph of a function $F:{mathbb{R}}^n{ ightarrow}{mathbb{R}}$ with |det DdF|=1, corresponds to a complete unimodular left symmetric algebra with a nondegenerate Hessian type inner product. We will investigate the condition for the domain over the homogeneous hypersurface to be homogeneous through an extension of the complete unimodular left symmetric algebra, which is called the graph extension.