  In this paper, we prove that, if a full irreducible infinite dimensional
anti-Kaehler isoparametric submanifold of codimension greater than one has
$J$-diagonalizable shape operators, then it is an orbit of the action of a
Banach Lie group generated by one-parameter transformation groups induced by
holomorphic Killing vector fields defined entirely on the ambient Hilbert
space.
