Determinant Factorization for Left Multiplication in the Sedenions

Abstract

We study zero-divisors in the $16$-dimensional sedenion algebra from the viewpoint of the determinant of left multiplication. We show that this determinant admits a canonical factorization into the square of a quartic polynomial, obtained via a $G_2$-invariant reduction to a quaternionic normal form and an explicit block computation. The quartic factor recovers the classical characterization of left zero-divisors in terms of the imaginary components. After normalization, the resulting zero-divisor manifold is identified with the Stiefel manifold $V_2(\mathbb{R}^7)$. We also analyze a $3$-dimensional purely imaginary slice, on which the quartic reduces to a simple quadratic form. This yields a concrete geometric model of the zero-divisor locus as a quadratic cone in the slice.