The first aim of this note is to fill a gap in the literature by proving that, given a global field K and a finite set S of primes of K, every finite split embedding problem G -> Gal(L/K) over K with nilpotent kernel has a solution Gal(F/K) -> G such that all primes in S are totally split in F/L. We then apply this to inverse Galois theory over division rings. Firstly, given a number field K of level at least 4, we show that every finite solvable group occurs as a Galois group over the division ring H-K of quaternions with coefficients in K. Secondly, given a finite split embedding problem with nilpotent kernel over a finite field K, we fully describe for which automorphisms sigma of K the embedding problem acquires a solution over the skew field of fractions K(T, sigma) of the twisted polynomial ring K[T, sigma].