First, must be investigated what this embedding means. Next, the existence must be proved for real number based Hilbert spaces. Subsequently, the prove must be given for complex number based Hilbert spaces and for quaternionic Hilbert spaces.

A model exists in which the embedding process is limited to a subspace of the quaternionic Hilbert space that scans over the whole separable Hilbert space.

Does this make the companion non-separable Hilbert space unique?