Special orthogonal sphere fibration ($SO(s)\to SO(s+1)\to S^s$)

The last-column map $p(A)=A{e}_{s+1}$ has fiber $SO(s)$, included as $\mathrm{diag}(B,1)$. Local completion of a unit vector by the Gram-Schmidt process gives local fiber bundle charts. Consequently it has the homotopy lifting property and the long exact sequence of homotopy groups of a fibration.

Project a based map $\alpha :S^r\to SO(s+1)$ to $S^s$. Since $r<s$, the resulting homotopy class vanishes. Lift a based null-homotopy by the special orthogonal sphere fibration. Its endpoint lies in the fiber $SO(s)$. If the lift moves its base point, right-multiply at time $t$ by the inverse of its base-point value, which lies in the fiber. This keeps the projected homotopy unchanged and makes the lift based. Thus $\alpha$ is based-homotopic to a map into $SO(s)$, proving the asserted surjectivity.