Part 10 of 15. The octonions give 7D the only cross product outside 3D. Their symmetry group is the exceptional Lie algebra G₂ = Der(𝕆). In 8 dimensions, vectors and spinors become interchangeable through triality. And Freudenthal's magic square arranges the rest of the exceptional family (F₄, E₆, E₇, E₈) from the same octonionic data via the exceptional Jordan algebra h₃(𝕆) and the Cayley plane 𝕆P².
The cross product $x \times y$ gives a vector perpendicular to both inputs. A true bilinear cross product exists only in dimensions 3 and 7.
These are the imaginary parts of the quaternions and octonions ($4-1=3$ and $8-1=7$). The formula is $x \times y = \text{Im}(xy)$, and the Fano plane is the multiplication table for the 7D cross product.
| x × y | -- |
| ⟨x, x×y⟩ | -- |
| ⟨y, x×y⟩ | -- |
Figure 1. The 7D cross product computed via octonion multiplication. Both dot products are zero, confirming perpendicularity — the defining property of a cross product.
The geometric rule: $x \times y = \text{Im}(xy)$. In the octonions, if you multiply two imaginary vectors, the result's imaginary part is exactly the 7D cross product. It obeys the Fano plane rules you learned in Part 9.
A symmetry of an algebra is a rotation preserving its multiplication: $(xy)' = x'y'$.
For complex numbers the symmetry is trivial; for quaternions it is $SO(3)$; for octonions it is G₂, the smallest exceptional Lie group.
An octonion basis permutation has to preserve the directed Fano triples, not just the unordered lines. Find one.
Figure 2. The Fano Lock. Most movements of the nodes will break the directed multiplication cycles of the Fano plane. The legal movements are product-preserving octonion symmetries.
The exceptional group: G₂ doesn't fit the standard families of rotations. It exists because of octonion non-associativity, and it is 14-dimensional.
G2 is the smallest exceptional simple Lie algebra. Its Dynkin diagram has two nodes connected by a triple edge. Two simple roots, 12 total roots, rank 2, dimension 14, Weyl group order 12 (dihedral). Cartan matrix:
with product A12 · A21 = (−1)(−3) = 3, confirming the triple edge, and with the asymmetry telling us the two roots have squared lengths in ratio 1 : 3 (α2 is long, α1 is short, arrow points from α2 to α1).
The 12 roots split into 6 short and 6 long. Short roots form a regular hexagon, long roots a larger hexagon rotated 30°. Overlaying them gives a star of David.
Figure 2b. The G2 root system: 6 short roots (blue) at inner-hexagon vertices plus 6 long roots (red) at outer-hexagon vertices. Together they form the "star of David" rosette with 12-fold dihedral symmetry. The Dynkin diagram on the right has two nodes joined by a triple edge with an arrow pointing from the long simple root α2 to the short simple root α1.
G2 has no matrix construction. The cleanest way to build it is as the derivation algebra of the octonions. A derivation D: 𝕆 → 𝕆 satisfies the Leibniz rule
Derivations form a Lie algebra under the commutator bracket [D1, D2] = D1D2 − D2D1. For ℝ there are no nonzero derivations; for ℂ and ℍ, the derivation algebras are 0 and 3-dimensional respectively. For the octonions, the derivation algebra is 14-dimensional, and it is G2:
Cartan identified G2 with Der(𝕆) in his 1894 thesis. Without the octonions the classification list would stop at A, B, C, D.
The smallest G2 representation is 7-dimensional, acting on Im(𝕆) ≅ ℝ7 by derivations. Since D(1) = 0, every derivation preserves Im(𝕆) and acts by a linear map preserving the octonion product. This is why the 7D cross product has symmetry group G2, and why G2-manifolds appear as exceptional cases in Berger's classification and in M-theory compactifications.
In 3D, vectors return to normal after a 360° rotation but spinors need 720°.
In 8D there is a three-way symmetry between vectors $V$, left-spinors $S_L$, and right-spinors $S_R$ — triality, the reason octonions bridge to superstring theory.
The triality rule: treating all three spaces as octonions, $v \times s_L = s_R$ is preserved. Rotating one space requires balancing the other two.
Figure 3. The Triality Balancer. In 8D, any rotation of space (V) can be perfectly offset by rotations of the particle states (S). This is the "magic of 8."
Physics hook: 10D spacetime (8 space + 2 time) uses triality between force (vectors) and matter (spinors) to support supersymmetry.
Triality shows up inside E8 one more time, and in a form you can count. Pick one of the 112 mixed-scalar octonion units $P \in E_8$, one of the roots with $\mathrm{Re}(P) = \pm\tfrac{1}{2}$ met in explainer 9. For any other root $Q$, the conjugation triad
produces two cousins $Q_2, Q_3$ that are again in $E_8$. Each triad is classified by the inner product $Q \cdot P$, and despite the octonions being non-associative, every triad satisfies $(Q_1 Q_2) Q_3 = Q_1 (Q_2 Q_3)$; the octonion product is associative on these three-element sets.
Figure 3b. The 78 triads generated by a fixed scalar-heavy root $P$, split by $Q \cdot P$. Positive and negative triads satisfy $Q_1 Q_2 Q_3 = \pm 1$; orthogonal triads close up without a scalar product. Counting by class: 18 + 18 + 42 = 78, which is exactly the dimension of the adjoint representation of $E_6$. The example panel shows a specific $Q$ for each class, with its Euclidean inner product with $P$ computed live.
Why this matters. Octonion non-associativity usually blocks group-like structures. Triad associativity is a local rescue: three octonions in an $E_8$-stabilised triad associate anyway. This is why $E_8$ appears wherever physicists need associativity-on-demand within an octonionic frame.
The Fano plane has three interpretations. Toggle between them to see how the seven points and lines encode the cross product table, the G₂-invariant product structure, and the quaternion-line structure.
Figure 4. The Unified Fano Lens. The same seven-point geometry encodes the 7D cross product (pick any two units, read off the third), the G2-invariant product structure (basis permutations that preserve every directed line), and the seven embedded copies of the quaternions (each Fano line is a quaternionic subalgebra of 𝕆).
Octonionic projective spaces stop at the plane. You can build an octonionic line () and the Cayley plane , but not : 3D projective space needs matrix multiplication, which requires associativity.
The 16-dimensional Cayley plane is the end of the projective ladder. Its isometry group is the 78-dimensional E6, with point stabiliser (52-dimensional). The next article meets both groups.
In the Cayley plane, "lines" are 8-dimensional spheres, and any two of these 8D lines must intersect at exactly one point.
Figure 5. The Cayley Plane Projection. A 2D slice of a 16D manifold. The intersection of two lines in is governed by the 52-dimensional symmetry of .
Hermitian matrices represent quantum observables, but products of Hermitian matrices are usually not Hermitian. Jordan's Jordan product averages orders: — commutative but non-associative.
Applied to octonionic Hermitian matrices, the Jordan product gives the 27-dimensional exceptional Jordan algebra (the Albert algebra): three real diagonal entries plus three octonionic off-diagonals. Its automorphism group is . Jordan, von Neumann, and Wigner (1934) found four infinite families of finite-dimensional Jordan algebras plus one exception: h3(𝕆).
Below is an incomplete octonionic matrix. For it to be a valid quantum observable, it must be Hermitian: entries below the diagonal must be the conjugates of the entries above.
| 5.0 | e1 + e4 | e7 - e2 |
| ? | 2.1 | e3 + e6 |
| ? | ? | -1.0 |
Figure 6. The Albert Algebra Workbench. This grid is the only exceptional Jordan algebra. Its 27 dimensions are the "matter" that acts upon by derivations.
The five exceptional Lie groups () seemed like accidents until Freudenthal and Tits (1950s–60s) found a construction that builds them all from pairs of normed division algebras: , where is the traceless part and Der the derivation algebra.
The table is symmetric: M(𝔸, 𝔹) = M(𝔹, 𝔸). The classical 3×3 block (ℝ, ℂ, ℍ only) gives 𝔰𝔬(3), 𝔰𝔲(3), 𝔰𝔭(3), 𝔰𝔲(6), 𝔰𝔬(12). The fourth row/column (where 𝕆 enters) gives F4, E6, E7, E8. G2 sits outside the square as Der(𝕆) — the undoubled construction.
The classical entries (top-left 3×3) use only reals, complexes, and quaternions. The exceptional entries (the entire octonion row and column) are where the five exceptional Lie groups live.
| ℝ | ℂ | ℍ | 𝕆 | |
|---|---|---|---|---|
| ℝ | SO(3) | SU(3) | Sp(3) | F₄ |
| ℂ | SU(3) | SU(3)×SU(3) | SU(6) | E₆ |
| ℍ | Sp(3) | SU(6) | SO(12) | E₇ |
| 𝕆 | F₄ | E₆ | E₇ | E₈ |
Figure 7. Freudenthal's Magic Square. Blue cells are classical groups; red cells involve the octonions and produce the exceptional Lie groups. The bottom-right corner, where octonions meet octonions, is , the ultimate 248-dimensional symmetry. The table is symmetric under swap of rows and columns, which is why F4, E6, E7 each appear twice.
The exceptions exist because the octonions exist. Octonion non-associativity produces F4, E6, E7, E8 through the magic square and G2 through Der(𝕆). Remove the octonions and the list becomes ABCD.
As each entry appears, the formula shows how Der(), , and Der() combine to produce the total dimension. Classical entries fill blue; exceptional entries fill red and gold.
| ℝ | ℂ | ℍ | 𝕆 | |
|---|---|---|---|---|
| ℝ | - | - | - | - |
| ℂ | - | - | - | - |
| ℍ | - | - | - | - |
| 𝕆 | - | - | - | - |
Figure 8. Build the magic square entry by entry. Each cell's dimension decomposes as Der(𝔸) + dim(𝔸₀ ⊗ 𝔹₀) + Der(𝔹).