  The nonzero level sets in $n$-dimensional flat affine space of a
translationally homogeneous function are improper affine spheres if and only if
the Hessian determinant of the function is equal to a nonzero constant multiple
of the $n$th power of the function. The exponentials of the characteristic
polynomials of certain left-symmetric algebras yield examples of such functions
whose level sets are analogues of the generalized Cayley hypersurface of
Eastwood-Ezhov. There are found purely algebraic conditions sufficient for the
characteristic polynomial of the left-symmetric algebra to have the desired
properties. Precisely, it suffices that the algebra has triangularizable left
multiplication operators and the trace of the right multiplication is a Koszul
form for which right multiplication by the dual idempotent is projection along
its kernel, which equals the derived Lie subalgebra of the left-symmetric
algebra.
