Flag manifold to Complex Projective line

Here's a picture:

First, view the picture "literally", as $\mathbf{R}^2$ equipped with Cartesian coordinates $(z_0, z_1)$. Every real line through the origin crosses at least one of the lines $V_1 = \{z_0=1\}$ or $V_0 = \{z_1=1\}$. (The $z_0$-axis misses $V_0$, the $z_1$-axis misses $V_1$.) Each "affine coordinate neighborhood" $V_i$ is a "copy" of $\mathbf{R}$, and the respective affine coordinates are related by $z_0z_1 = 1$.

The circle of unit vectors in $\mathbf{R}^2$ is denoted $S$. Each real line through the origin cuts $S$ in a unit $0$-sphere, i.e., in a pair of antipodal points (labeled $U$).

Further, each real line hits the circle labeled $\mathbf{P}^1$ exactly twice: once at the origin, and once at an arbitrary point of the circle. (The $z_1$-axis is tangent to the small circle, i.e., "hits the circle twice at the origin".) Consequently, the real projective line may be viewed as the small circle. Projection away from the origin is stereographic projection from $\mathbf{P}^1$ to the affine coordinate neighborhood $V_1$.

Now view the picture "figuratively", as $\mathbf{C}^2$, and think of complex lines through the origin. The affine coordinate neighborhoods are defined exactly as before. Each is a "copy" of $\mathbf{C}$ in $\mathbf{C}^2$, and except for the axes, every complex line through the origin hits each affine coordinate neighborhood exactly once.

The set $S$ of unit vectors is the $3$-sphere $S^3$, and each complex line through the origin intersects $S$ in a unit circle $U$. (These circles are the "fibres of the Hopf map".)

In this interpretation, $\mathbf{P}^1$ is not sitting in $\mathbf{C}^2$ in a nice way; however, it may still be viewed abstractly as the result of attaching the two complex lines $V_0$ and $V_1$ by identifying $z_1$ and $1/z_0$, so the complex projective line is a real $2$-sphere.

So, finally, the original question boils down to: The group $U(2)$ of unitary $2 \times 2$ matrices acts transitively on the space of complex lines through the origin in $\mathbf{C}^2$, and the stabilizer of a line is isomorphic to a real $2$-torus. (Each coordinate axis is invariant by the diagonal subgroup, for example.)