  It is conjectured that any trigonometric Olshanetsky-Perelomov Hamiltonian
written in Fundamental Trigonometric Invariants (FTI) as coordinates takes an
algebraic form and preserves an infinite flag of spaces of polynomials. It is
shown that try-and-guess variables which led to the algebraic form of
trigonometric Olshanetsky-Perelomov Hamiltonians related to root spaces of the
classical $A_N, B_N, C_N, D_N, BC_N$ and exceptional $G_2, F_4$ Lie algebras
are FTI. This conjecture is also confirmed for the trigonometric $E_6$
Olshanetsky-Perelomov Hamiltonian whose algebraic form is found with the use of
FTI.
