Geodesic
A $\mathbb Z_4^3$-grading on a $56$-dimensional simple structurable algebra and related fine gradings on the simple Lie algebras of type $E$
Commentationes Mathematicae Universitatis Carolinae, Tome 55 (2014) no. 3, pp. 285-313

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We describe two constructions of a certain $\mathbb Z_4^3$-grading on the so-called Brown algebra (a simple structurable algebra of dimension $56$ and skew-dimension $1$) over an algebraically closed field of characteristic different from $2$. The Weyl group of this grading is computed. We also show how this grading gives rise to several interesting fine gradings on exceptional simple Lie algebras of types $E_6$, $E_7$ and $E_8$.
Classification : 17A30, 17B25, 17B70, 17C40
Keywords: graded algebra; structurable algebra; exceptional simple Lie algebra 
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     title = {A $\mathbb Z_4^3$-grading on a $56$-dimensional simple structurable algebra and related fine gradings on the simple {Lie} algebras of type $E$},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
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Aranda-Orna, Diego; Elduque, Alberto; Kochetov, Mikhail. A $\mathbb Z_4^3$-grading on a $56$-dimensional simple structurable algebra and related fine gradings on the simple Lie algebras of type $E$. Commentationes Mathematicae Universitatis Carolinae, Tome 55 (2014) no. 3, pp. 285-313. http://geodesic.mathdoc.fr/item/CMUC_2014__55_3_a2/