I think I have found what I needed: the usual Double Centralizer Theorem (as say in Herstein, "Noncommutative Rings, Th. 4.3.2, or Rowen, "Ring Theory - Student Edition", Th. 7.1.9) does not apply, as indicated, due to lack of finite-dimensionality over the centre. However, one can apply Rieffel's (Double Centralizer) Theorem, as say found in Lam, "A First Course in Noncommutative Rings", Th. 3.11, or Lang "Algebra 3.ed", Th. XVII.5.4, by taking the small skew field $D$ as the simple ring resp. the large skew field of the extension as the non-zero module involved. St.

I think I may have found something useful: the usual Double Centralizer Theorem (as say in Herstein, "Noncommutative Rings, Th. 4.3.2, or Rowen, "Ring Theory - Student Edition", Th. 7.1.9) does not apply, as indicated, due to lack of finite-dimensionality over the centre. However, Rieffel's (Double Centralizer) Theorem, as say found in Lam, "A First Course in Noncommutative Rings", Th. 3.11, or Lang "Algebra 3.ed", Th. XVII.5.4, is nearer the mark (by taking the small skew field $D$ as the simple ring involved), but the large skew field of the extension is not an ideal of $D$ ... something is still missing, I'm afraid ... St.