The only linear associative algebra in which the coordinates are real numbers and products vanish only if one factor is zero are the field of real numbers, the field of complex numbers, and the algebra of quaternions with real coefficients.
Peirce's Theorem
See also
Complex Number, Quaternion, Real Number, Weierstrass's TheoremExplore with Wolfram|Alpha
References
Schafer, R. D. "The Peirce Decomposition." §3.2 in An Introduction to Nonassociative Algebras. New York: Dover, pp. 32-37, 1996.Referenced on Wolfram|Alpha
Peirce's TheoremCite this as:
Weisstein, Eric W. "Peirce's Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PeircesTheorem.html