For u ∉ Q̄ with uū algebraic and z ∉ Q̄ + Q̄u + Q̄ū, the space Q̄ + Q̄u + Q̄ū + Q̄z carries a 2×3 configuration exactly when z ∈ Q̄ + Q̄u + Q̄ū + Q̄w for w = u², ū² or 1/(u − a), a ∈ Q̄ non-zero

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For u ∉ Q̄ with uū algebraic and z ∉ Q̄ + Q̄u + Q̄ū, the space Q̄ + Q̄u + Q̄ū + Q̄z carries a 2×3 configuration exactly when z ∈ Q̄ + Q̄u + Q̄ū + Q̄w for w = u², ū² or 1/(u − a), a ∈ Q̄ non-zero · Prove2Me