  The compact exceptional Lie groups F4, E6, E7 and E8 have spinor groups as a
subgroup as follows: E8 \supset Ss(16) \supset Spin(15) \supset Spin(14)
\supset Spin(13), E7 \supset Spin(12) \supset Spin(11), E6 \supset Spin(10), F4
\supset Spin(9) \supset Spin(8) \supset Spin(7) \supset \cdot \cdot \cdot
\supset Spin(1) \ni 1. We know the involution {\sigma}' induced an element
{\sigma}' \in Spin(8) \subset F4 \subset E6 \subset E7 \subset E8. Now, in this
paper, we determine the group structures of (Spin(n)){\sigma}' which are the
fixed subgroups by the involution {\sigma}'.
