Fix a basis $(e^{\prime}_{1},\ldots,e^{\prime}_{r})$ of $V$. For any $\alpha\in(0,1)$, there exists an $\alpha$-orthogonal basis $(e_{1},\ldots,e_{r})$ of $V$ with respect to $\|\raisebox{1.72218pt}{.}\|$ such that $(e_{1},\ldots,e_{r})$ is compatible with $(e^{\prime}_{1},\ldots,e^{\prime}_{r})$. Moreover, if the absolute value $|\raisebox{1.72218pt}{.}|$ is discrete, then there exists an orthogonal basis $(e_{1},\ldots,e_{r})$ of $V$ compatible with $(e^{\prime}_{1},\ldots,e^{\prime}_{r})$.