for "a" a commutative ring, one can define a determinant det: gl("a") → "a*" to the group of units of "a", which vanishes on e("a") and thus descends to a map det: "k(a)" → "a*". as e("a") ◅ sl("a"), one can also define the special whitehead group sk("a") := sl("a")/e("a"). this map splits via the map "a*" → gl(1, "a") → "k(a)" (unit in the upper left corner), and hence is onto, and has the special whitehead group as kernel, yielding the split short exact sequence: