Grover iteration in a nonorthogonal basis ($G=\left(\begin{array}{cc}1& 2a\\ -2a& 1-4a^2\end{array}\right)$)

For normalized marked and uniform states $|w⟩,|s⟩$ with real overlap $a\in (0,1)$, the two reflections $(2|s⟩⟨s|-I)(I-2|w⟩⟨w|)$ have the displayed matrix in that ordered nonorthogonal basis. The coordinate Gram matrix is $S=\left(\begin{array}{cc}1& a\\ a& 1\end{array}\right)$ and $G^{†}SG=S$. With $a=\sin t$, its eigenvalues are $e^{\pm 2it}$. Applied $r$ times to $|s⟩$, it gives marked-state success ${\sin }^2((2r+1)t)$. In these coordinates the overlap with the marked state is the first coefficient plus $a$ times the second; it is not generally the first coefficient alone.