  Four level quantum systems, known as quartits, and their relation to two-
qubit systems are investigated group theoretically. Following the spirit of
Klein's lectures on the icosahedron and their relation to Hopf sphere bra-
tions, invariants of complex re ection groups occuring in the theory of qubits
and quartits are displayed. Then, real gates over octits leading to the Weyl
group of E8 and its invariants are derived. Even multilevel systems are of
interest in the context of solid state nuclear magnetic resonance.
