4 added 153 characters in body

Personally I like the definition in Barton, Sudbery paper (thank you, Bruce for adding the reference):

MR2020553 (2005b:17017) Barton, C. H. ; Sudbery, A. Magic squares and matrix models of Lie algebras. Adv. Math. 180 (2003), no. 2, 596--647.

This is also available at: http://arxiv.org/abs/math/0203010

It uses triality algebra based on R, C, H, O composition algebras. Using this I have constructed all compact and non-compact exceptional Lie algebras in GAP.

Magic square correspond to square of algebras:
R*R, R*C, R*H, R*O
C*R, C*C, C*H, C*O
H*R, H*C, H*H, H*O
O*R, O*C, O*H, O*O
where * is the tensor product. You can replace algebra A with split version {A^~} to obtain non compact version.

Lie algebra in position A*B is TriA + TriB + A*B + A*B + A*B. What is remaining is just to define the bracket. To obtain f4 with compact spin9 I have changed sign in last two A*B.

Regards, Marek

3 more details; added 25 characters in body

Personally I like the definition in Barton, Sudbery paper (thank you, Bruce for adding the reference):

MR2020553 (2005b:17017) Barton, C. H. ; Sudbery, A. Magic squares and matrix models of Lie algebras. Adv. Math. 180 (2003), no. 2, 596--647.

This is also available at: http://arxiv.org/abs/math/0203010

It uses triality algebra based on R, C, H, O composition algebras. Using this I have constructed all compact and non-compact exceptional Lie algebras in GAP.

I have not yet reached

Magic square correspond to square of algebras:
R*R, R*C, R*H, R*O
C*R, C*C, C*H, C*O
H*R, H*C, H*H, H*O
O*R, O*C, O*H, O*O
where * is the 19-th century papers on this subjecttensor product.That must be interesting

Lie algebra in position A*B is TriA + TriB + A*B + A*B + A*B. What is remaining is just to define the bracket.

Regards, Marek

2 Edited to include reference

Personally I like the definition in Barton, Sudbery paper:

MR2020553 (you can find exact reference in Baez)2005b:17017) Barton, C. H. ; Sudbery, A. Magic squares and matrix models of Lie algebras. Adv. Math. 180 (2003), no. 2, 596--647.

This is also available at: http://arxiv.org/abs/math/0203010

It uses triality algebra based on R, C, H, O composition algebras. Using this I have constructed all compact and non-compact exceptional Lie algebras in GAP.

I have not yet reached the 19-th century papers on this subject. That must be interesting.

Regards, Marek

1