Linear models of the exceptional Lie algebra ${\mathfrak{e}}_8$

This work provides five explicit constructions of the exceptional Lie algebra ${\mathfrak{e}}_8$, based on its semisimple subalgebras of maximal rank. Each of these models is graded by an abelian group, namely, ${\mathbb{Z}}_4$, ${\mathbb{Z}}_5$, ${\mathbb{Z}}_6$, ${\mathbb{Z}}_3^2$ and ${\mathbb{Z}}_2\times {\mathbb{Z}}_4$; the neutral component is direct sum of special linear algebras and the remaining homogeneous components are irreducible modules for it.