ROTATIONS IN SPACE OF EVEN DIMENSIONS. 161 



2. Invariant planes. If (2) has a real root X, there must be a 

 real vector r such that 



/ = (/)•/• = \ri 



Since r' and r have the same length 



X= ± 1. 



In case of a proper motion, there is a second real root 



1 



In case of an improper motion the second root is 



1 



In either case there are two real roots (equal for a proper motion). 

 There is then an invariant plane. That there is an invariant plane 

 also follows from the fact that the line along which a lies is left invari- 

 ant point for point and consequently the space of 2n — 1 dimensions 

 perpendicular to this is left invariant. In this space the character- 

 istic equation is of odd order and so has a real root. Two vectors are 

 thus left invariant and so their plane is invariant. In this plane each 

 point is left fixed. 



If (2) has no real root it must have two conjugate imaginary roots. 

 Two conjugate imaginary vectors are therefore left invariant. The 

 real plane containing these two vectors is then left invariant. Hence 

 in any case a rotation in 2n dimensions has an invariant real plane. 

 This plane is left invariant as a plane but the individual points are not 

 left invariant. Since the 2/i — 2 dimensional space perpendicular 

 to this invariant plane is also left invariant, by an application of the 

 same argument there is an invariant plane in this perpendicular space. 

 By continuing we see that the rotation must leave n completely perpen- 

 dicular real planes invariant}^ 



13 It follows from this fact that positive rotations can be defined. For 

 take two perpendicular unit vectors in each invariant plane arranged so that 

 the rotation in each plane is from the first to the second. Then a positive 

 rotation can be defined as one such that the outer product of these unit vectors 

 (keeping the order indicated in each plane) is positive. 



