I was shown the following in my text on Krylov subspaces but I don't know the steps taken. Given $ \mathbf{Ax} = \mathbf{b} $ and $ \mathbf{A} \in\mathbb{R}^{n\times n}$ is non-singular and $\mathbf{A}$ is symmetric, with respect to the oblique projection framework, i.e. $W_k = \mathbf{A}K_k$: $$ \langle \mathbf{r}_0, \mathbf{r}_k\rangle = \|\mathbf{r}_k \|^2$$
$\mathbf{r}$ is the residual for each iteration, $\mathbf{r}_k = \mathbf{b} - \mathbf{Ax}_k$. I have the following but don't really know where I am going: $$\begin{aligned} \mathbf{r}_0^T \mathbf{r}_k &= (\mathbf{b} - \mathbf{Ax}_0)^T(\mathbf{b} - \mathbf{Ax}_k) \\ &= (\mathbf{b}^T - \mathbf{x}_0^T\mathbf{A}^T)(\mathbf{b} - \mathbf{Ax}_k) \\ &= \mathbf{b}^T\mathbf{b} - \mathbf{b}^T\mathbf{Ax}_k - \mathbf{x}_0^T\mathbf{A}^T\mathbf{b} + \mathbf{x}_0^T\mathbf{A}^T\mathbf{A}\mathbf{x}_k \\ &= z + \mathbf{x}_0^T\mathbf{A}^T\mathbf{A}\mathbf{x}_k \end{aligned}$$
$$\begin{aligned} \text{RHS} &= \| \mathbf{r}_k \|^2 \\ &= \mathbf{r}_k^T \mathbf{r}_k \\ &= (\mathbf{b} - \mathbf{Ax}_k)^T(\mathbf{b} - \mathbf{Ax}_k) \\ &= (\mathbf{b}^T - \mathbf{x}_k^T\mathbf{A}^T)(\mathbf{b} - \mathbf{Ax}_k) \\ &= \mathbf{b}^T\mathbf{b} - \mathbf{b}^T\mathbf{Ax}_k - \mathbf{x}_k^T\mathbf{A}^T\mathbf{b} + \mathbf{x}_k^T\mathbf{A}^T\mathbf{Ax}_k \\ &= \mathbf{b}^T\mathbf{b} - \mathbf{b}^T\mathbf{Ax}_k - (\mathbf{x}_0 + \sum_{i=0}^{k-1} \alpha_i\mathbf{A}^i\mathbf{r}_0)^T\mathbf{A}^T\mathbf{b} + (\mathbf{x}_0 + \sum_{i=0}^{k-1} \alpha_i\mathbf{A}^i\mathbf{r}_0)^T\mathbf{A}^T\mathbf{A}\mathbf{x}_k \\ &= \mathbf{b}^T\mathbf{b} - \mathbf{b}^T\mathbf{Ax}_k - \mathbf{x}_0^T\mathbf{A}^T\mathbf{b} - (\sum_{i=0}^{k-1} \alpha_i\mathbf{A}^i\mathbf{r}_0)^T\mathbf{A}^T\mathbf{b} + \mathbf{x}_0^T\mathbf{A}^T\mathbf{A}\mathbf{x}_k + (\sum_{i=0}^{k-1} \alpha_i\mathbf{A}^i\mathbf{r}_0)^T\mathbf{A}^T\mathbf{A}\mathbf{x}_k \\ &=z - (\sum_{i=0}^{k-1} \alpha_i\mathbf{A}^i\mathbf{r}_0)^T\mathbf{A}^T\mathbf{b} + \mathbf{x}_0^T\mathbf{A}^T\mathbf{A}\mathbf{x}_k + (\sum_{i=0}^{k-1} \alpha_i\mathbf{A}^i\mathbf{r}_0)^T\mathbf{A}^T\mathbf{A}\mathbf{x}_k \\ &= z + \mathbf{x}_0^T\mathbf{A}^T\mathbf{A}\mathbf{x}_k + (\sum_{i=0}^{k-1} \alpha_i\mathbf{A}^i\mathbf{r}_0)^T\mathbf{A}^T\mathbf{A}\mathbf{x}_k - (\sum_{i=0}^{k-1} \alpha_i\mathbf{A}^i\mathbf{r}_0)^T\mathbf{A}^T\mathbf{b} \\ &= z + \mathbf{x}_0^T\mathbf{A}^T\mathbf{A}\mathbf{x}_k + (\sum_{i=0}^{k-1} \alpha_i\mathbf{A}^i\mathbf{r}_0)^T\mathbf{A}^T\mathbf{A}(\mathbf{x}_0 + (\sum_{i=0}^{k-1} \alpha_i\mathbf{A}^i\mathbf{r}_0)) - (\sum_{i=0}^{k-1} \alpha_i\mathbf{A}^i\mathbf{r}_0)^T\mathbf{A}^T(\mathbf{r}_0 + \mathbf{Ax}_0) \\ &= z + \mathbf{x}_0^T\mathbf{A}^T\mathbf{A}\mathbf{x}_k + (\sum_{i=0}^{k-1} \alpha_i\mathbf{A}^i\mathbf{r}_0)^T\mathbf{A}^T\mathbf{A}(\sum_{i=0}^{k-1} \alpha_i\mathbf{A}^i\mathbf{r}_0) - (\sum_{i=0}^{k-1} \alpha_i\mathbf{A}^i\mathbf{r}_0)^T\mathbf{A}^T\mathbf{r}_0 \\ &= z + \mathbf{x}_0^T\mathbf{A}^T\mathbf{A}\mathbf{x}_k + (\sum_{i=0}^{k-1} \alpha_i\mathbf{A}^i\mathbf{r}_0)^T\mathbf{A}^T(\mathbf{A}(\sum_{i=0}^{k-1} \alpha_i\mathbf{A}^i\mathbf{r}_0) - \mathbf{r}_0) \end{aligned}$$
Not really sure where to go from here. I believe I have to do something with oblique projection but don't know what.