  The automorphism groups of the 27 lines on the smooth cubic surface or the 28
bitangents to the general quartic plane curve are well-known to be closely
related to the Weyl groups of $E\_6$ and $E\_7$. We show how classical
subconfigurations of lines, such as double-sixes, triple systems or Steiner
sets, are easily constructed from certain models of the exceptional Lie
algebras. For ${\mathfrak e}\_7$ and ${\mathfrak e}\_8$ we are lead to
beautiful models graded over the octonions, which display these algebras as
plane projective geometries of subalgebras. We also interpret the group of the
bitangents as a group of transformations of the triangles in the Fano plane,
and show how this allows to realize the isomorphism $PSL(3,F\_2)\simeq
PSL(2,F\_7)$ in terms of harmonic cubes.
