Abstract. Let Q be a Dynkin quiver of type E with exceptional vertex y and Q' the quiver obtained from Q by deleting y. Let M be the maximal indecomposable representation of Q. The restriction of M to Q' turns out to be the direct sum of three pairwise orthogonal indecomposable representations A(1), A(2), A(3). These representations seem to play a decisive role in the representation theory of the Dynkin quiver Q and of the corresponding Euclidean quiver. The lecture will provide a characterization of this antichain triple. Note that the reduction scheme follows the rules of the magic Freudenthal-Tits square - thus it relates the exceptional Dynkin quivers to the Euclidean Hurwitz algebras (the complex numbers, the quaternions and the octonions). We will discuss in which way the octonions may be reconstructed using the representation theory of E6.