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 homogeneous.
"Homogeneity of Infinite Dimensional Anti-Kaehler Isoparametric Submanifolds." Tokyo J. Math. 37 (1) 159 - 178, June 2014. https://doi.org/10.3836/tjm/1406552436