matsumoto's original theorem is even more general: for any root system, it gives a presentation for the unstable k-theory. this presentation is different from the one given here only for symplectic root systems. for non-symplectic root systems, the unstable second k-group with respect to the root system is exactly the stable k-group for gl("a"). unstable second k-groups (in this context) are defined by taking the kernel of the universal central extension of the chevalley group of universal type for a given root system. this construction yields the kernel of the steinberg extension for the root systems "a" ("n" > 1) and, in the limit, stable second "k"-groups.