Considering $\mathbb{C}$ as an algebra over $\mathbb{R}$, it is easy to find a vector space isomorphism between $\mathbb{C}\otimes\mathbb{C}$ and $\mathbb{C}\oplus\mathbb{C}$. I am struggling to justify why $f:\mathbb{C}\otimes\mathbb{C}\rightarrow\mathbb{C}\oplus\mathbb{C}$, determined by the following: \begin{equation} 1\otimes1\mapsto(1,1) \end{equation} \begin{equation} 1\otimes i\mapsto(i,-i) \end{equation} \begin{equation} i\otimes1\mapsto(i,i) \end{equation} \begin{equation} i\otimes i\mapsto(-1,1) \end{equation}
determines an algebra isomorphism. My notes say that 'the multiplication table of $\mathbb{C}\otimes\mathbb{C}$ is preserved', but I'm not really sure what this means. I'm aware that there is a similar question here, but the answer given uses methods beyond my level, and I guess my question is more specific to the above mapping. Thanks in advance!