TITLE: What's the minimal embedding of orthogonal Grassmannian?
QUESTION [5 upvotes]: Suppose $X$ is the orthogonal Grassmanian.  We know the Plücker embedding does not span the whole background $\mathbb{CP}^n,$ just a subspace $\mathbb{CP}^m.$  

My question:  is there an expression of the isometric embedding of $X$ into $\mathbb{CP}^m$?  

By expression, I mean a way to write down explicitly the coordinates such as Plücker coordinates for the Grassmannian?

REPLY [3 votes]: I stumbled over here looking for something else. 
I think the short answer is to construct the orthogonal Grassmannian of isotropic n-planes in an 2n-dimensional space, take a list of all the principal pfaffians of a skew-symmetric n by n matrix. Since odd-pfaffians automatically vanish, the construction is slightly different in the even and odd cases. 
In general there is a construction of the minimal rational embedding of each compact hermitian symmetric space. In these cases the variety is the unique closed orbit of a highest weight vector and the minimal rational embedding is into the span of the orbit of the highest weight vector. For a complete picture see  Landsberg and Manivel's work https://arxiv.org/pdf/math/9902102.pdf