In this paper, we define the notion of the complex Coxeter group associated with a proper complex equifocal submanifold in a symmetric space of non-compact type. We prove that a proper complex equifocal submanifold is decomposed into a non-trivial (extrinsic) product of two such submanifolds if and only if its associated complex Coxeter group is decomposable. Its proof is performed by showing a splitting theorem for an infinite-dimensional proper anti-Kaehlerian isoparametric submanifold.
"A splitting theorem for proper complex equifocal submanifolds." Tohoku Math. J. (2) 58 (3) 393 - 417, 2006. https://doi.org/10.2748/tmj/1163775137