Paper 1 - Note on the Restriction of G2 Invariants to SU (3)

We record a simple algebraic observation relating the invariant theory of the exceptional Lie algebra \(G_2\) to that of \(SU(3)\). Identifying \(\mathfrak{su}(3) \subset \mathfrak{g}_2\) with the long-root subalgebra of \(G_2\), the standard basic invariants satisfy \[ I_6^{(G_2)} = 2\bigl(I_3^{(A_2)}\bigr)^2 - \bigl(I_2^{(A_2)}\bigr)^3 \] on the common Cartan subalgebra. This induces a map \(\iota(a,c) = (a, 2c^2 - a^3)\) from the \(SU(3)\) orbit space to the \(G_2\) orbit space, which is a bijection when restricted to the half-chamber \(c\ge 0\), and a diffeomorphism between interiors. The factor of two appearing throughout is exactly the index \([W(G_2):W(A_2)] = 2\) of the corresponding Weyl groups. We also record the associated pullback of the Hessian metric. This note is purely algebraic and makes no claims about physical applications.