  The Kochen-Specker theorem states that a 3-dimensional complex Euclidean
space admits a finite configuration of projective lines such that the
corresponding quantum observables (the orthogonal projectors) cannot be
assigned with 0 and 1 values in a classically consistent way. This paper shows
that the irreducible root systems of exceptional and of non-crystallographic
types are useful in constructing such configurations in other dimensions. The
cases $E_6$ and $E_7$ lead to new examples, while $F_4$, $E_8$, and $H_4$,
yield a new interpretation of the known ones. The described configurations have
an additional property: they are saturated, i.e. the tuples of mutually
orthogonal lines, being partially ordered by inclusion, yield a poset with all
maximal elements having the same cardinality (the dimension of space).
