Let $\mathcal{T}_{n}(\underline{r})$ be a $k$-Tate alegbra. Let $I\subseteq\{1,\dots,n\}$ be a subset of indices such that $\{\alpha(\log r_{i})\}_{i\in I}$ are $\mathbb{Q}$-linearly independent and $|I|$ is maximal for this independence property. Let $\boldsymbol{r}_{I}=(r_{i_{1}},\dots,r_{i_{1}})$, then $K_{\boldsymbol{r}_{I}}\widehat{\otimes}_{k}\mathcal{T}_{n}(\boldsymbol{r})$ is a strict $K_{\boldsymbol{r}_{I}}$-Tate algebra.