Consider an embedding $$\Phi: G_{k_1}(R^{n_1})\times G_{k_2}(R^{n_2})\rightarrow G_k(R^n)$$ of the product of two Grassmannians $G_{k_1}(R^{n_1})\times G_{k_2}(R^{n_2})$ into $G_k(R^n)$, where $G_k(R^n)$ denotes the Grassmannian of k-plane in $R^n$.

Assume that the dimension of the product $G_{k_1}(R^{n_1})\times G_{k_2}(R^{n_2})$ plus $1$ is equal to the dimension of $G_k(R^n)$, namely $$k_1(n_1-k_1)+k_2(n_2-k_2)+1=k(n-k).$$

QUESTION: Is it true that the only possibilities for the existence of such embedding are:

either $\Phi: P^1\times P^1\rightarrow P^3$

or $\Phi: P^{n-1}\rightarrow P^n$, where $P^{n-1}=G_1(R^n)=G_{n-1}(R^n)$ denote the real projective space?