0027

There are exactly twenty-seven straight lines on a smooth cubic surface,which give a basis of the fundamental representation of Lie algebra E 6 {\displaystyle \mathrm {E_{6}} }. The unique simple formally real Jordan algebra, the exceptional Jordan algebra of self-adjoint 3 by 3 matrices of quaternions, is 27-dimensional;[5] its automorphism group is the 52-dimensional exceptional Lie algebra F 4 . {\displaystyle \mathrm {F_{4}} .}