For a real connected simple noncompact Lie group \(G\) with Iwasawa decomposition \(G=KAN\) the structure of the subgroup \(M=Z(A)\cap K\) is known in case \(G\) is a linear group. The subgroup \(B=MAN\) is a minimal parabolic subgroup of \(G\) and since \(A\) is a vector group and \(N\) is simply connected nilpotent, the topological structure of \(B\) is closely linked to that of \(M\). In the paper under review the author provides a description of the group \(M\) for any connected, simply connected and nonlinear simple Lie group \(G\). The paper contains a short overview about Clifford algebras and spinors, and representations of the subgroup \(D_n = \{ e_{i_{1}}\cdot\ldots\cdot e_{i_{2l}}\,| \, 1\leq i_{1},\ldots,i_{2l} \leq n\}\) of Spin(\(n\)), which make up the major ingredients of the proofs. Most of the effort for determining the structure of \(M\) is spent with a case by case study of the exceptional groups.