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Are there algebras over reals besides complex numbers, where identities, analoguous to $(-1)...

I will address the question in the case $f(z) = (-1)^z = \exp(z \log(-1))$ and $A$ is a finite-dimensional commutative (associative unital) $\mathbb{R}$-algebra, where $\log(-1)$ is a suitable choice …
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