Ambient Lipschitz equivalence of real surface singularities

We present a series of examples of pairs of singular semialgebraic surfaces (real semialgebraic sets of dimension two) in ${\mathbb{R}}^3$ and ${\mathbb R}^4$ which are bi-Lipschitz equivalent with respect to the outer metric, ambient topologically equivalent, but not ambient Lipschitz equivalent. For each singular semialgebraic surface $S\subset {\mathbb{R}}^4$, we construct infinitely many semialgebraic surfaces which are bi-Lipschitz equivalent with respect to the outer metric, ambient topologically equivalent to $S$, but pairwise ambient Lipschitz non-equivalent.