I've been reading a paper and came across the following inequality $$- \langle B^T B y^k, x^{k+1} - x^k\rangle \leq \frac{1}{2} \left(\|x^{k}-x^{k-1}\|_{B^T B}^2-\|x^{k+1}-x^{k}\|_{B^T B}^2 - \|y^{k}\|_{B^T B}^2\right)$$ where $y^{k}=(x^{k+1}-x^{k})-(x^{k}-x^{k-1})$.
Now instead of $B^T B$, I have a matrix $A$. If $A$ is symmetric positive definite, using Cholesky decomposition, I can get to the same previous expression but I have a matrix $A$ that is not symmetric and have the following form $$ A = \begin{bmatrix} 0_d & I_d & 0_d & \dots & 0_d & 0_d & 0_d \\ 0_d & 0_d & 0_d & \dots & 0_d & 0_d & 0_d\\ 0_d & I_d & 0_d & I_d & \dots & 0_d & 0_d\\ 0_d & 0_d & 0_d & 0_d & \dots & 0_d & 0_d\\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ 0_d & 0_d & 0_d & \dots & I_d & 0_d & I_d\\ 0_d & 0_d & 0_d & \dots & 0_d & 0_d & 0_d \end{bmatrix} $$ where the even rows of $A$ are equal to the vector whose elements are all zeros, and $0_d$ and $I_d$ denotes the zeros matrix and the identity matrix of order $d$, respectively.
I was trying to derive an upper bound that have a similar form (a bound that is a function of the three terms $\|x^{k}-x^{k-1}\|^2$, $-\|x^{k+1}-x^{k}\|^2$, and $- \|y^{k}\|^2$. Currently, the bound I have is $$- \langle A y^k, x^{k+1} - x^k\rangle \leq \frac{1}{2} \left(\|A y^k - (x^{k+1}-x^{k})\|^2-\|x^{k+1}-x^{k}\|^2 - \|A y^{k}\|^2\right)$$ and I was trying to express the first term $\|A y^k - (x^{k+1}-x^{k})\|^2$ as a function of the terms $\|x^{k}-x^{k-1}\|^2$ and $-\|x^{k+1}-x^{k}\|^2$ but didn't get through.