Given a non-singular quadratic form q of maximal Witt index on V := V (2n + 1, F), let Δ be the building of type B n formed by the subspaces of V totally singular for q and, for 1k . In this paper we give a new very easy proof of this fact. We also prove that if char(F) = 2 then dim(As a consequence, when 1 < k < n and char(F) = 2 the embedding ε k is not universal. Finally, we prove that if F is a perfect field of characteristic p > 2 or a number field, n > k and k = 2 or 3, then ε k is universal.