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In $\mathbb{R}^n$, given an unit $L_1$ sphere $\mathcal{B}_n: |x_1|+|x_2|+\ldots+|x_n|\leq 1$ and a hyperplane $\mathcal{P}: a_1x_1+a_2x_2+\ldots+a_nx_n=0$. Does there always exist a rotation such that $\mathcal{B}\cap\mathcal{P}$ is embedded into the $(n-1)$ dimensional unit $L_1$ sphere: $\mathcal{B}_{n-1}: |x_1|+|x_2|+\ldots+|x_n-1|\leq 1, x_n=0$ after the rotation?

In $\mathbb{R}^n$, given an unit $L_1$ sphere $\mathcal{B}_n: |x_1|+|x_2|+\ldots+|x_n|\leq 1$ and a hyperplane $\mathcal{P}: a_1x_1+a_2x_2+\ldots+a_nx_n=0$. Does there always exist a rotation such that $\mathcal{B}_n\cap\mathcal{P}$ is embedded into the $(n-1)$ dimensional unit $L_1$ sphere: $\mathcal{B}_{n-1}: |x_1|+|x_2|+\ldots+|x_{n-1}|\leq 1, x_n=0$ after the rotation?