= Solution
For the compact <alpha-plane> $A_\ell$ and <beta plane> $B_H$, an intersection point must satisfy $\ell\subset W\subset H$. If $\ell\not\subset H$, that is impossible. If $\ell\subset H$, intermediate two-planes correspond to lines $W/\ell$ in the two-dimensional quotient $H/\ell$. Therefore
$$
\boxed{A_\ell\cap B_H=\begin{cases}\varnothing,&\ell\not\subset H,\\\mathbb P(H/\ell)\cong\mathbb{CP}^1,&\ell\subset H.\end{cases}}
$$
The nonempty intersection is a projective <null geodesic>. In an affine chart its tangent must be both $u^A\pi^{A'}$ and $\lambda^A v^{A'}$; these conditions force a multiple of $\lambda^A\pi^{A'}$. Thus its affine portion is a null line, with the indicated projective compactification.
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