Filtrations in semisimple Lie algebras, II

In this paper, we continue our study of the maximal bounded $\mathbb{Z}$-filtrations of a complex semisimple Lie algebra $L$. Specifically, we discuss the functionals which give rise to such filtrations, and we show that they are related to certain semisimple subalgebras of $L$ of full rank. In this way, we determine the ``order'' of these functionals and count them without the aid of computer computations. The main results here involve the Lie algebras of type $E_6$, $E_7$ and $E_8$, since we already know a good deal about the functionals for the remaining types. Nevertheless, we reinterpret our previous results into the new context considered here. Finally, we describe the associated graded Lie algebras of all of the maximal filtrations obtained in this manner.