BQ-Pauli Multiplication Table

The BQ-Pauli algebra has eight 2×2 complex basis matrices: the space-time basis T̂, Î, Ĵ, K̂ (minquat) and the spin-norm basis 1̂, σ̂_I, σ̂_J, σ̂_K (pauliquat), with T̂ = i1̂ and K̂_i = T̂σ̂_i. Choose a cell: the row is the left factor, the column the right factor. The page multiplies the two matrices entry by entry and identifies the outcome as a signed basis element.

Choose a product

Left right Write Î, Ĵ, K̂ as K̂_I, K̂_J, K̂_K

result in the space-time basis T̂, K̂_i result in the spin-norm basis 1̂, σ̂_i selected reversed order

The explicit multiplication

Reversed order

The eight basis matrices

Matrices as defined in the BQ-Pauli slides (de Haas, 24 September 2026). Every product on this page is computed live from these entries; nothing is looked up except the check against the paper's table.

What the table shows

Squares. Every space-time element squares to −1̂ (T̂² = Î² = Ĵ² = K̂² = −1̂); every spin-norm element squares to +1̂.
Quaternion products. ÎĴ = −ĴÎ = K̂, ĴK̂ = −K̂Ĵ = Î, K̂Î = −ÎK̂ = Ĵ: the spatial K̂_i anticommute and close on themselves.
Pauli products. σ̂_Iσ̂_J = −K̂, σ̂_Jσ̂_K = −Î, σ̂_Kσ̂_I = −Ĵ: two spin-norm elements give a space-time element, with a minus sign in cyclic order.
Mixed products. K̂_iσ̂_i = σ̂_iK̂_i = T̂ on the diagonal; off the diagonal K̂_iσ̂_j = −σ̂_jK̂_i lands on ±σ̂_k.
T̂ is central. T̂ = i1̂ commutes with everything: T̂σ̂_i = K̂_i and T̂K̂_i = −σ̂_i. Multiplying by T̂ moves an element between the two bases.

Reading the colours

Because T̂ = i1̂ and K̂_i = iσ̂_i, each space-time element is i times its spin-norm partner. Which basis a product lands in depends on the pair:

space-time · space-time: K̂_iK̂_j (i ≠ j) stays in the space-time basis; the squares give −1̂ and T̂K̂_i gives −σ̂_i, both in the spin-norm basis;
spin-norm · spin-norm: σ̂_iσ̂_j (i ≠ j) gives ∓K̂_k in the space-time basis; the squares give +1̂;
mixed: T̂σ̂_i = K̂_i and K̂_iσ̂_i = T̂ land in the space-time basis, K̂_iσ̂_j (i ≠ j) in the spin-norm basis.

So the space-time basis alone does not close under multiplication: T̂T̂ = −1̂ and T̂K̂_i = −σ̂_i leave it, and the spin-norm basis is needed to span the full biquaternion algebra.

Method, as in the reproduction appendix: use the explicit matrices, execute the multiplication, decompose the result on the basis, and compare with the stated table. The green pill confirms the computed outcome equals the entry in the paper's table. E.P.J. de Haas, BQ algebra. Part of the BQ algebra models; see also bq-galaxy-models.