The geometric interpretations of all real exceptional simple Lie groups of classes $G_2$, $F_4$, $E_6$, $E_7$ and $E_8$ are described. In particular, we describe the interpretations of the four last classes as groups of motions of elliptic and hyperbolic planes over algebras of octaves and split octaves and over tensor products of them and algebras of usual and split complex numbers, quaternions and octaves. The explicite expressions of motions of these planes are found. The symmetry figures and parabolic figures of all considered spaces and geometric interpretations of all fundamental linear representations of real exceptional simple Lie groups are found.