 More precisely, there are two objects called Cayley planes, namely the real and the complex Cayley plane.The real Cayley plane is the symmetric space F4/Spin(9), where F4 is a compact form of an exceptional Lie group and Spin(9) is the spin group of nine-dimensional Euclidean space (realized in F4).It admits a cell decomposition into three cells, of dimensions 0, 8 and 16.