The complexified tautological line resolves real-projective K-theory extensions

The transition functions of a real line bundle take values in $\{\pm 1\}$, so the transition functions of $\lambda {\otimes }_{\mathbb{R}}\lambda$ are squares of $\pm 1$, hence equal to $1$, and $\lambda {\otimes }_{\mathbb{R}}\lambda$ is trivial.

The $K$-AHSS of ${\mathbb{R}\mathbb{P}}^r$ has nonzero entries only in even coefficient rows. A differential $d_s$ with $s$ even has odd target row and vanishes. Let $s$ be odd and let its source column satisfy $p\ge 1$. If $p=r$, as can occur in the top integral column when $r$ is odd, then the target column $r+s$ is outside the complex. If $0<p<r$, a nonzero source has $p$ even by [A3], so its target column $p+s$ is odd and the target vanishes unless $p+s=r$; in that remaining case the source is a torsion group while $H^r({\mathbb{R}\mathbb{P}}^r;\mathbb{Z})=\mathbb{Z}$ is torsion-free, so the differential vanishes as well. Finally, a differential with source in the $0$-column cannot be hit, and the $0$-column edge identifies $E_{\mathrm{\infty }}^{0,q}\cong h^q(\mathrm{p}\mathrm{t})=\mathbb{Z}=E_2^{0,q}$, so those differentials vanish too. Hence all differentials vanish and $E_2=E_{\mathrm{\infty }}$, so the associated graded of $K^0({\mathbb{R}\mathbb{P}}^r)$ consists of the integral cohomology of the projective space in even degrees, with the degree-two class in filtration two.

Complexifying the triviality of the transition functions gives $\xi {\otimes }_{\mathbb{C}}\xi \cong {\epsilon }_{\mathbb{C}}^1$; writing $\alpha =[\xi ]-1$ in the ring of [A2] therefore gives $(1+\alpha {)}^2=1$, that is ${\alpha }^2=-2\alpha$, and multiplying repeatedly gives ${\alpha }^k=(-2{)}^{k-1}\alpha$ for every $k\ge 1$.

If $m=0$, both ${\mathbb{R}\mathbb{P}}^0$ and ${\mathbb{R}\mathbb{P}}^1\cong S^1$ have trivial reduced $K^0$ by [A5], so $\alpha =0$ has the asserted order $1$. Now suppose $m\ge 1$. On ${\mathbb{R}\mathbb{P}}^{2m}$, [A5] gives ${\alpha }^m\mathrm{\ne }0$ and ${\alpha }^{m+1}=0$. On ${\mathbb{R}\mathbb{P}}^{2m+1}$, the restriction isomorphism of [A5] carries the tautological $\alpha$ to the even-dimensional one, so the same two power statements hold there as well. With ${\alpha }^2=-2\alpha$ from step 2.1, consequently $2^m\alpha =(-1{)}^m{\alpha }^{m+1}=0$ while $2^{m-1}\alpha =(-1{)}^{m-1}{\alpha }^m\mathrm{\ne }0$. Thus $\alpha$ has exact order $2^m$ and generates ${\tilde{K}}^0({\mathbb{R}\mathbb{P}}^r)$; the two $K^1$ calculations are the final clauses of [A5].

Steps 1.2, 2.1 and 3.1 give the asserted relations, the exact order of $\alpha$, the cyclicity of the reduced group and the two odd-group computations.