The adjoint group of a simple complex Lie algebra g has a unique minimal orbit in the projective space P g , 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 C ¯ of the union kC of all spaces spanned by k points on C. The image of k C ¯ in P g is usually called the ( k − 1 ) st secant variety of P C . These higher secant varieties are known, and easily determined, for g = sl n or g = sp 2 n ; for completeness, we give short proofs of these results. Our main contribution is therefore the explicit description of k C ¯ for g = o n , where the embedding of P C into P o n is isomorphic to the Plücker embedding of the Grassmannian of isotropic lines in P n − 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 2 n , the sets kC are not all closed in o n , and we present a partial result on the nilpotent orbits contained in them.