We deal with two forms of the "uniqueness cases" in the classification of large simple K*-groups of finite Morley rank of odd type, where large means the 2-rank m2 is at least three. This substantially extends results known for even larger groups having Prüfer 2-rank at least three, so as to cover the two groups PSp 4 and G 2. With an eye towards more distant developments, we carry out this analysis for L*-groups, a context which is substantially broader than (...) the K* setting. (shrink)