  In this paper we provide novel results on the infinite level normal form and
orbital normal form classifications of nonlinear Eulerian and rotational vector
fields with two pairs of non-resonant imaginary modes. We use the method of
multiple Lie brackets and its extension along with time rescaling for orbital
normal form classification. Furthermore, we apply two reduction techniques. The
first is to use the radical Lie ideal of rotational vector fields and its
corresponding quotient Lie algebra. The second technique is to employ a Schur
complement block matrix type in Gaussian elimination and analysis of block
matrices. The infinite level parametric normal form classification are also
presented. The latter is also viewed as a normal form result for multiple-input
controlled systems with non-resonant double Hopf singularity. We also discuss
nonlinear symmetry transformations associated with the nonlinear symmetry group
of the simplest normal forms. Symbolic normal form transformation generators
are derived for computer algebra implementation. Further, the results are
efficiently implemented and verified using Maple for all three types of normal
form computations up to arbitrary degree, where they can also include both
small bifurcation parameters and arbitrary symbolic constant coefficients.
