  We develop an approach in solving exactly the problem of three-body
oscillators including general quadratic interactions in the coordinates for
arbitrary masses and couplings. We introduce a unitary transformation of three
independent angles to end up with a diagonalized Hamiltonian. Using the
representation theory of the group $SU(3)$, we explicitly determine the
solutions of the energy spectrum. Considering the ground state together with
reduced density matrix, we derive the corresponding purity function that is
giving rise to minimal and maximal entanglement under suitable conditions. The
cases of realizing one variable among three is discussed and know results in
literature are recovered.
