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Complex slices, inverses, and deformations of a three-dimensional real algebra
We study the real three-dimensional unital algebra with basis $\{1,i,j\}$ and products $i^2=j^2=-1,\quad ij=0,\quad ji=i+j$. Its quadratic and power-associative elements form the union of exactly three complex subalgebras; their nonzero elements are precisely those with a two-sided inverse, which is
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