G is a group, \(\sigma\) an automorphism of G, \(G^{\sigma}=\{g\in G |\sigma g=g\}\). In this paper the automorphisms \(\sigma\) of order 3 are found and \(G^{\sigma}\) is realized for simply connected compact exceptional Lie groups \(G=G_ 2\), \(F_ 4\) and \(E_ 6\) where \(G_ 2\) is an automorphism group of the Cayley division algebra, \(F_ 4\) is an automorphism group of the Jordan algebra and \(E_ 6\) is an automorphism group of the complex exceptional Jordan algebra.