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
