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An octonionic construction of $E_8$ and the Lie algebra magic square

Robert A. Wilson, Tevian Dray and Corinne A. Manogue

Vol. 20 (2023), No. 2-3, 611–634
Abstract

We give a new construction of the Lie algebra of type E8, in terms of 3 × 3 matrices, such that the Lie bracket has a natural description as the matrix commutator. This leads to a new interpretation of the Freudenthal–Tits magic square of Lie algebras, acting on themselves by commutation.

Dedicated to the memory of Jacques Tits

Keywords
exceptional Lie algebras, Freudenthal–Tits magic square, octonions
Mathematical Subject Classification
Primary: 20G41
Milestones
Received: 11 April 2022
Revised: 26 September 2022
Accepted: 26 October 2022
Published: 13 September 2023
Authors
Robert A. Wilson
School of Mathematical Sciences
Queen Mary University of London
London
United Kingdom
Tevian Dray
Department of Mathematics
Oregon State University
Corvallis, OR
United States
Corinne A. Manogue
Department of Physics
Oregon State University
Corvallis, OR
United States