The authors make use of the well-known relationship between the exceptional subalgebras of \(E_ 8\) and octonions to build, for the adjoint and the lowest-dimensional representations of the exceptional algebras, the vector spaces, corresponding to their maximal subalgebras. They construct the vector spaces, corresponding to the exceptional subalgebras of \(E_ 8\) and the way, exceptional algebras embed each other, also the vector spaces, corresponding to some particular classical subalgebras of \(E_ 8\), including the maximal ones, and finally, describe and classify the behaviour of the adjoint and fundamental representations of \(E_ 7\), \(E_ 6\) and \(F_ 4\) with respect to their maximal classical subalgebras.