C01.2 · Derivation

Automorphisms fix the unit and preserve the norm

If \(g\) is an invertible multiplicative linear map, \(g(1)\) is a two-sided identity on the image, hence is \(1\). Each element obeys \[ x^2-t(x)x+n(x)1=0,\qquad t(x)=x+\bar x. \] For nonscalar \(x\), the coefficients of \(x\) and \(1\) are unique. Apply \(g\) to this equation and compare the quadratic equation for \(g(x)\): both trace and norm are preserved. Scalars are already fixed. This argument uses the composition-algebra quadratic identity, not a positive-definite spectral theorem.

Likewise \(D(1)=D(1)1+1D(1)\) gives \(D(1)=0\). Exponentiating a derivation yields automorphisms in the identity component: differentiating \(e^{-tD}((e^{tD}x)(e^{tD}y))\) gives zero by the Leibniz rule. This argument does not classify possible disconnected components.

Solve the derivation equations rather than count imagined generators

Bilinearity makes the basis-pair system sufficient for all real \(x,y\). The executed system has rank and nullity \[ (50,14) \] for both products (`split.derivation_rank_nullity`, `compact.derivation_rank_nullity`). Each nullspace basis is checked back in the full defining system. Every ordered commutator is also expressed in that basis, establishing closure of the computed space.

Let \(K(D,E)=\operatorname{tr}(\operatorname{ad}D\,\operatorname{ad}E)\), the **intrinsic Killing form**, not an unlabelled ambient restriction. Constructing the adjoint matrices gives \[ K(D,E)=4\operatorname{tr}_{8}(DE) \] on both computed algebras. Its exact inertia, in the order positive/negative/zero, is \[ (8,6,0)\quad\hbox{for the split product},\qquad (0,14,0)\quad\hbox{for the compact product}. \] These are the corresponding `Killing_inertia` outputs. No numerical eigenvalue cutoff enters their calculation.

For the split product, \(\theta(D)=-D^T\) preserves the derivation space and squares to the identity. The form \[ -K(D,\theta D)=4\operatorname{tr}(DD^T) \] is positive for nonzero real \(D\). The computed fixed and anti-fixed spaces therefore give the Cartan decomposition with dimensions \[ \dim\mathfrak k=6,\qquad\dim\mathfrak p=8 \] (`split.Cartan_dimensions(k,p)`). On the compact product every derivation is skew in the positive Euclidean metric.

The split root data are computed, not inferred from the dimension

In Zorn coordinates, take the commuting derivations \[ \delta_H(a,u,v,b)=(0,Hu,-H^Tv,0) \] with \(H=\operatorname{diag}(1,-1,0)\) and \(H=\operatorname{diag}(0,1,-1)\). Their joint adjoint weights, including multiplicities, are printed in full in `split.Cartan_joint_weights`. They span the whole computed derivation algebra: a zero-weight space of dimension \(2\) and twelve nonzero one-dimensional weight spaces. The intrinsic Cartan Killing matrix is \[ \begin{pmatrix}16&-8\\-8&16\end{pmatrix}. \] The root squared lengths are \(1/12\) and \(1/4\), each with multiplicity \(6\), so the length ratio is \(3\). These are the root data conventionally called \(G_2\), with a real split Cartan. This is substantially more information than the isolated dimension \(14\).

**Global boundary:** exhaustion of the full nonlinear automorphism group, its connected components and a bibliographic proof of its precise global identification are **[NOT VERIFIED]** in this packet. The source's \(G_{2(2)}\) name is consistent with the computed split Lie algebra; the article does not use dimension alone as a global classification proof.

The quaternion compact action and its central kernel

For unit quaternions \(a,b\), define \[ F_{a,b}(p,q)=(apa^{-1},\,bqa^{-1}). \] Quaternion associativity, \(\overline{uv}=\bar v\bar u\), and unit norms show directly that this preserves the stated split product. For example, the cross term in the transformed first half is \[ \overline{bsa^{-1}}\,bqa^{-1} =a\bar s\,b^{-1}bq\,a^{-1}=a\bar s q\,a^{-1}. \] The other terms transform in the same specified manner. Thus this is an action of the product of the two unit-quaternion groups, not just a matching dimension.

If \(F_{a,b}\) is the identity, conjugation by \(a\) fixes every quaternion, so \(a\) is a central unit, \(a=\pm1\). Evaluating the second half at \(q=1\) then gives \(b=a\). The kernel is exactly \(\{(1,1),(-1,-1)\}\). Hence the faithful compact family is the quotient by this diagonal central sign, not the direct product asserted without qualification in the March prompt. The finite quaternion-unit sample checks the product law, but the all-unit kernel conclusion follows from this algebraic argument, not from sample exhaustion.

On the quaternion \(q\)-space this is the norm-preserving action \(q\mapsto bqa^{-1}\). Its image is connected and has the dimension \(6\) computed for the compact Lie algebra; the finite kernel does not lower dimension. It therefore fills the connected orientation-preserving orthogonal group of that four-dimensional Euclidean space. In conventional notation the faithful compact family is \[ \mathrm{SO}(4)\cong \bigl(\mathrm{SU}(2)\times\mathrm{SU}(2)\bigr)/ \{(1,1),(-1,-1)\}. \] This identifies the explicitly constructed compact family. It is not an exhaustion proof for the full nonlinear split-octonion automorphism group.

The Lie-algebra compact-part obstruction carried by LIB2-096 now has a concrete construction behind its dimension comparison. It excludes an injective copy of an eight-dimensional compact colour algebra **in that six-dimensional compact part**. It does not select a compact colour real form elsewhere.