[en] The projective representations of the rotation subgroup W+(Φ) have been determined from those of W(Φ) for each root system Φ. This is done by constructing non-trivial central extensions of W+(Φ) via the double coverings of the rotation groups SO(l). This adaptation gives a unified way of obtaining the basic projective representations of W+(Φ) from those of W(Φ). In particular, formulae giving irreducible characters of these representations are explicitly determined in each case. (author). 19 refs, 2 tabs