The new construction given by Barton and Sudbery of the Freudenthal-Tits magic square, which includes the exceptional classical simple Lie algebras, will be interpreted and extended by using a pair of symmetric composition algebras, instead of the standard unital composition algebras.
"The Magic Square and Symmetric Compositions." Rev. Mat. Iberoamericana 20 (2) 475 - 491, June, 2004.