Consider a finite-dimensional, associative algebra presented as follows: $\mathcal{A} = e_1 \mathbb{R}\oplus e_2 \mathbb{R} \oplus \cdots \oplus e_n \mathbb{R} $ with multiplication $*: \mathcal{A} \times \mathcal{A} \rightarrow \mathcal{A}$ defined by linearly extending $ \color{red}{ e_i *e_j = C_{ij}^ke_k \qquad \text{no summation over $k$ intended}.} $ For example, the direct product algebra $e_i*e_j = \delta_{ij}e_i$ fits this condition.
The clarity of the multiplication is partly from a good choice of notation. If I took a different basis then the multiplication could get muddled. For example, $\mathbb{R}^2$ with the commutative direct product algebra is described by $e_1 *e_1=e_1$ and $e_2*e_2=e_2$ and $e_1*e_2=0$. If I instead presented the multiplication by $f_1 = 2e_1+3e_2$ and $f_2=e_1+e_2$ then $f_1*f_1 = 5f_1-6f_2$.
Apparently I could be given an algebra where a proper change of notation would induce the $\color{red}{\text{desired property}}$. For the sake of this question, let us define $\mathcal{A}$ as desirable iff there exists some basis $\{f_1,\dots f_n \}$ for $\mathcal{A}$ such that $f_i *f_j = C_{ij}^kf_k$ where no summation is intended over $k$.
Question: is there a more standard term for a desirable algebra?
Please point me towards a respected reference for your term if at all possible. Thanks is advance for your help!