Grassmann and Stiefel manifolds as affine varieties

Let $\mathbb K = \mathbb R$, $\mathbb C$ or $\mathbb H$, the skew $\mathbb R$-algebra of quaternions.
Recall that the Grassmann manifold $G_{n,r}(\mathbb K)$ is defined as a set of all $r$-dimensional linear subspaces of the space $\mathbb K ^n$ and the Stiefel manifold $V_{n,r}(\mathbb K)$ as the set of all orthonormal $r$-frames in $\mathbb K^n$.
We aim to examine the structure of Grassmann and Stiefel manifolds via their presentations as algebraic sets.
The main result is:
Theorem. (1) The tangent bundle $T G_{n,r}(\mathbb K) = \{(A, B) \in G_{n,r}(\mathbb K) \times M_n(\mathbb K); \overline{B}t = B, AB + BA = B\}$.
(2) There is an algebraic isomorphism $T G_{n,r}(\mathbb K) \approx \mathrm{Idem}_{r,n}(\mathbb K)$, where $\mathrm{Idem}_{r,n}(\mathbb K) = \{A \in M_n(\mathbb K); A^2 = A, \mathrm{rk}(A) = r\}$.
(3) the $\mathbb C$-Zariski closure $\overline{G_{n,r}(\mathbb C)} = T G_{n,r}(\mathbb C)$ for $G_{n,r}(\mathbb C)$ as an $\mathbb R$-affine variety.
The joint project with Professor F.R. Gomez, Malaga, Spain.