The inverse of the tetraquaternion, M 4 (R)
Abstract
The real tetraquaternion algebra T = H ⊗ H, of dimension sixteen, associative and isomorphic to M 4 (R) = Cl(3, 1), is not a division algebra: the product of an element by its conjugate is not a scalar, and no norm gives the inverse. We compute the unique inverse of an element R = A + BI + CJ + DK, four real quaternions on the second set of units, by the quasideterminants of Gelfand and Retakh applied to the 4 × 4 quaternionic matrix of the linear system RR ′ = 1: the four quaternions of the inverse are the inverses of the four quasideterminants of the first row, expressions of 82 terms and inversion height four, and the left and right inverses so obtained coincide. The nested quaternion inverses collapse to a closed form, equivalent to the involution formulas known for Clifford algebras of dimension four but written on the four quaternions and made fully explicit: four cubic quaternion numerators over a single real quartic r
where the G are the six Gram determinants of the pairs and det(A, B, C, D) the volume of the four quaternions; r is the determinant of the 4 × 4 real representation, and the sixteen real components of the inverse are given explicitly, thirty-one cubic terms each over the 196 monomials of r. The inverse exists if and only if r ̸ = 0; r is multiplicative, of degree four, indefinite, and vanishes on a fifteen-dimensional cone of zero divisors; the inverse is a cubic polynomial in R; the inverse of a pure element is pure only when Tr R 3 = 0. Examples are worked out, and the algebra is compared with R, C, H and O. For the rotation operator Q → RQR -1 of the 15D Lorentz transformation, the conjugate is the inverse only on the compact subgroup SO(4); the nine non-compact directions, the boosts among them, need the true inverse, and for r = 1 the sixteen cubic numerators alone give it. The same sixteen expressions hold verbatim for the complex tetraquaternion M 4 (C), and for the element of u(2, 2) of a spinless body r = u 2 + (1 + 6D)u -8D 3 , with the action D = x • p and u = (x • x)(p • p) in units of ℏ and ℏ 2 : it vanishes on a hypersurface which, at large actions, is the light cone of the origin of the frame and, at actions of order ℏ, the Compton sphere and Compton time of the body.
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BibTeX
@misc{reynaud2026inverse,
title = {{The inverse of the tetraquaternion, M 4 (R)}},
author = {Patrick .B Reynaud},
howpublished = {HAL},
year = {2026},
month = sep,
url = {https://hal.science/hal-05749774v2},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The inverse of the tetraquaternion, M 4 (R)
AU - Patrick .B Reynaud
PY - 2026
DA - 2026/09/14
PB - HAL
UR - https://hal.science/hal-05749774v2
AB - The real tetraquaternion algebra T = H ⊗ H, of dimension sixteen, associative and isomorphic to M 4 (R) = Cl(3, 1), is not a division algebra: the product of an element by its conjugate is not a scalar, and no norm gives the inverse. We compute the unique inverse of an element R = A + BI + CJ + DK, four real quaternions on the second set of units, by the quasideterminants of Gelfand and Retakh applied to the 4 × 4 quaternionic matrix of the linear system RR ′ = 1: the four quaternions of the inverse are the inverses of the four quasideterminants of the first row, expressions of 82 terms and inversion height four, and the left and right inverses so obtained coincide. The nested quaternion inverses collapse to a closed form, equivalent to the involution formulas known for Clifford algebras of dimension four but written on the four quaternions and made fully explicit: four cubic quaternion numerators over a single real quartic r where the G are the six Gram determinants of the pairs and det(A, B, C, D) the volume of the four quaternions; r is the determinant of the 4 × 4 real representation, and the sixteen real components of the inverse are given explicitly, thirty-one cubic terms each over the 196 monomials of r. The inverse exists if and only if r ̸ = 0; r is multiplicative, of degree four, indefinite, and vanishes on a fifteen-dimensional cone of zero divisors; the inverse is a cubic polynomial in R; the inverse of a pure element is pure only when Tr R 3 = 0. Examples are worked out, and the algebra is compared with R, C, H and O. For the rotation operator Q → RQR -1 of the 15D Lorentz transformation, the conjugate is the inverse only on the compact subgroup SO(4); the nine non-compact directions, the boosts among them, need the true inverse, and for r = 1 the sixteen cubic numerators alone give it. The same sixteen expressions hold verbatim for the complex tetraquaternion M 4 (C), and for the element of u(2, 2) of a spinless body r = u 2 + (1 + 6D)u -8D 3 , with the action D = x • p and u = (x • x)(p • p) in units of ℏ and ℏ 2 : it vanishes on a hypersurface which, at large actions, is the light cone of the origin of the frame and, at actions of order ℏ, the Compton sphere and Compton time of the body.
ER -