  We noticed a discrepancy between \'Elie Cartan and Sigurdur Helgason about
the lowest possible dimension in which the simple exceptional Lie group ${\bf
E}_8$ can be realized. This raised the question about the lowest dimensions in
which various real forms of the exceptional groups ${\bf E}_\ell$ can be
realized. Cartan claims that ${\bf E}_6$ can be realized in dimension 16.
However Cartan refers to the complex group ${\bf E}_6$, or its split real form
$E_I$. His claim is also valid in the case of the real form denoted by
$E_{IV}$. We find however that the real forms $E_{II}$ and $E_{III}$ of ${\bf
E}_6$ can not be realized in dimension 16 \`a la Cartan. In this paper we
realize them in dimension 24 as groups of CR automorphisms of certain CR
structures of higher codimension.
  As a byproduct of these two realizations, we provide a full list of CR
structures $(M,H,J)$ and their CR embeddings in an appropriate ${\bf C}^N$,
which satisfy the following conditions:
  (1) they have real codimension $k>1$,
  (2) the real vector distribution $H$ proper for the action of the complex
structure $J$ is such that $[H,H]+H=TM$,
  (3) the local group $G_J$ of CR automorphisms of the structure $(M,H,J)$ is
simple, acts transitively on $M$ and has isotropy $P$ being a parabolic
subgroup in $G_J$,
  (4) the local symmetry group $G$ of the vector distribution $H$ on $M$
coincides with the group $G_J$ of CR automorphisms of $(M,H,J)$. Because all
the CR structures from our list satisfy the last property we call them
accidental. Our CR structures of higher codimension with the exceptional
symmetries $E_{II}$ and $E_{III}$ are particular entries in this list.
