The adjoint group of a simple complex Lie algebra g has a unique minimal orbit in the projective space Pg, whose pre-image in g we denote by C. We explicitly describe, for every classical g and every natural number k, the Zariski closure (k) over bar(C) over bar of the union kC of all spaces spanned by k points on C. The image of (k) over bar(C) over bar in Pg is usually called the (k - 1)st secant variety of PC. These higher secant varieties are known, and easily determined, for g = sl(n) or g = sp(2n); for completeness, we give short proofs of these results. Our main contribution is therefore the explicit description of (k) over bar(C) over bar for g = o(n), where the embedding of PC into Po-n is isomorphic to the Phicker embedding of the Grassmannian of isotropic lines in Pn-1 into p((n+1)n/2-1). We show that the first and the second secant variety are then characterised by certain conditions on the eigenvalues of matrices in o(n), while the third and higher secant varieties coincide with those of the Grassmannian of all projective lines. Finally, unlike for g = sl(n) or sp(2n), the sets kC are not all closed in o(n), and we present a partial result on the nilpotent orbits contained in them. (C) 2004 Elsevier Inc. All rights reserved.