Subgroups of the Spinor Group that Contain a Split Maximal Torus. II.

In the first paper of the series, we proved the standardness of a subgroup H containing a split maximal torus in the split spinor group Spin<MATH>(n,R)</MATH> over a field K of characteristic different from 2 containing at least 7 elements under one of the following additional assumptions: (1) H is reducible, (2) H is imprimitive, (3) H contains a nontrivial root element. In the present paper, we complete the proof of a result announced by the author in 1990 and prove the standardness of all intermediate subgroups, provided that <MATH>n=2l</MATH> and <MATH>|K|\ge 9</MATH>. For an algebraically closed K, this follows from a classical result of Borel and Tits, and for a finite K this was proved by Seitz. Similar results for subgroups of the orthogonal groups SO<MATH>(n,R)</MATH> were previously obtained by the author not only for fields, but for any commutative semilocal rings R with residue fields large enough. Bibliography: 52 titles. [ABSTRACT FROM AUTHOR]