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.
"Homogeneity of Infinite Dimensional Anti-Kaehler Isoparametric Submanifolds II." Tokyo J. Math. 40 (2) 301 - 337, December 2017. https://doi.org/10.3836/tjm/1502179231