Explicit diffeomorphism between an infinite dimensional sphere its product with itself

Question

Let $S$ be an infinite dimensional sphere in a Hillbert space.

As $S$ is homotopic to the product $S \times S$, then $S$ is diffeomorphic to $S \times S$ (for Hilbert manifolds, a homotopy equivalence implies diffeomorphism).

Is there an explicit diffeomorphism between $S$ and $S \times S$?