  Algebraic number theory relates SIC-POVMs in dimension $d>3$ to those in
dimension $d(d-2)$. We define a SIC in dimension $d(d-2)$ to be aligned to a
SIC in dimension $d$ if and only if the squares of the overlap phases in
dimension $d$ appear as a subset of the overlap phases in dimension $d(d-2)$ in
a specified way. We give 19 (mostly numerical) examples of aligned SICs. We
conjecture that given any SIC in dimension $d$ there exists an aligned SIC in
dimension $d(d-2)$. In all our examples the aligned SIC has lower dimensional
equiangular tight frames embedded in it. If $d$ is odd so that a natural tensor
product structure exists, we prove that the individual vectors in the aligned
SIC have a very special entanglement structure, and the existence of the
embedded tight frames follows as a theorem. If $d-2$ is an odd prime number we
prove that a complete set of mutually unbiased bases can be obtained by
reducing an aligned SIC to this dimension.
