diff --git a/Wanted.lean b/Wanted.lean
index 3f1c677a341d4e..77e6fbb39c4e81 100644
--- a/Wanted.lean
+++ b/Wanted.lean
@@ -10,6 +10,7 @@ public import Wanted.Geometry.Euclidean.Volume.Measure
 public import Wanted.Geometry.Manifold.PoincareConjecture
 public import Wanted.Geometry.Manifold.SmoothEmbedding
 public import Wanted.GroupTheory.GroupAction.Jordan
+public import Wanted.LinearAlgebra.RootSystem.Classification
 public import Wanted.Order.KrullDimension
 public import Wanted.Probability.Combinatorics.BinomialRandomGraph.Defs
 public import Wanted.Probability.Distributions.Binomial
diff --git a/Wanted/LinearAlgebra/RootSystem/Classification.lean b/Wanted/LinearAlgebra/RootSystem/Classification.lean
new file mode 100644
index 00000000000000..8c99b7cfc92398
--- /dev/null
+++ b/Wanted/LinearAlgebra/RootSystem/Classification.lean
@@ -0,0 +1,122 @@
+/-
+Copyright (c) 2026 Oliver Nash. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Oliver Nash
+-/
+module
+
+public import Mathlib.LinearAlgebra.Matrix.Cartan
+public import Mathlib.LinearAlgebra.RootSystem.CartanMatrix
+public import Mathlib.LinearAlgebra.RootSystem.OfBilinear
+
+/-!
+# The classification of root systems.
+
+-/
+
+namespace RootPairing
+
+variable
+  {ι : Type*} [Finite ι]
+  {K : Type*} [Field K] [CharZero K]
+  {M : Type*} [AddCommGroup M] [Module K M]
+  {N : Type*} [AddCommGroup N] [Module K N]
+  {P : RootPairing ι K M N}
+
+def Base.HasCM {n : Type*} [P.IsCrystallographic] (bs : P.Base) (A : Matrix n n ℤ) :=
+  ∃ e, bs.cartanMatrix.reindex e e = A
+
+/-! *Uniqueness*
+
+Note that we already have `RootPairing.Base.equivOfCartanMatrixEq` so the statement below about
+Cartan matrices really is all that is required. Note also that the triple-bond case is essentially
+`RootPairing.IsG2.card_base_support_eq_two` (+ related API).
+
+-/
+
+theorem_wanted Base.hasCM_A_or_B_or_C_or_D_or_E_or_F_or_G
+    [P.IsReduced] [P.IsCrystallographic] [P.IsIrreducible] [P.IsRootSystem]
+    (bs : P.Base) :
+    (∃ n, bs.HasCM (CartanMatrix.A n)) ∨
+    (∃ n, bs.HasCM (CartanMatrix.B n)) ∨
+    (∃ n, bs.HasCM (CartanMatrix.C n)) ∨
+    (∃ n, bs.HasCM (CartanMatrix.D n)) ∨
+    bs.HasCM (CartanMatrix.E 6) ∨
+    bs.HasCM (CartanMatrix.E 7) ∨
+    bs.HasCM (CartanMatrix.E 8) ∨
+    bs.HasCM CartanMatrix.F₄ ∨
+    bs.HasCM CartanMatrix.G₂
+
+/-! *Existence*
+
+Probably the best route is to construct these by developing further API for `RootPairing.ofBilinear`
+and then invoking it with appropriate matrices over `ℤ`.
+
+-/
+
+variable (ι K M N) (n : ℕ) [NeZero n]
+
+def_wanted a : RootPairing (Fin <| n * (n + 1)) K M N
+instance_wanted : (❰a❱ K M N n).IsReduced
+instance_wanted : (❰a❱ K M N n).IsIrreducible
+instance_wanted : (❰a❱ K M N n).IsValuedIn ℤ
+instance_wanted : (❰a❱ K M N n).IsRootSystem
+theorem_wanted a_hasCM_a : ∀ bs : (❰a❱ K M N n).Base, bs.HasCM (CartanMatrix.A n)
+
+def_wanted b : RootPairing (Fin <| 2 * n * n) K M N
+instance_wanted : (❰b❱ K M N n).IsReduced
+instance_wanted : (❰b❱ K M N n).IsIrreducible
+instance_wanted : (❰b❱ K M N n).IsValuedIn ℤ
+instance_wanted : (❰b❱ K M N n).IsRootSystem
+theorem_wanted b_hasCM_b : ∀ bs : (❰b❱ K M N n).Base, bs.HasCM (CartanMatrix.B n)
+
+def_wanted c : RootPairing (Fin <| 2 * n * n) K M N
+instance_wanted : (❰c❱ K M N n).IsReduced
+instance_wanted : (❰c❱ K M N n).IsIrreducible
+instance_wanted : (❰c❱ K M N n).IsValuedIn ℤ
+instance_wanted : (❰c❱ K M N n).IsRootSystem
+theorem_wanted c_hasCM_c : ∀ bs : (❰c❱ K M N n).Base, bs.HasCM (CartanMatrix.C n)
+
+def_wanted d : RootPairing (Fin <| 2 * n * (n - 1)) K M N
+instance_wanted : (❰d❱ K M N n).IsReduced
+theorem_wanted d_isIrreducible (hn : n ≠ 2) : (❰d❱ K M N n).IsIrreducible
+instance_wanted : (❰d❱ K M N n).IsValuedIn ℤ
+instance_wanted : (❰d❱ K M N n).IsRootSystem
+theorem_wanted d_hasCM_d : ∀ bs : (❰d❱ K M N n).Base, bs.HasCM (CartanMatrix.D n)
+
+def_wanted e₆ : RootPairing (Fin 72) K M N
+instance_wanted : (❰e₆❱ K M N).IsReduced
+instance_wanted : (❰e₆❱ K M N).IsIrreducible
+instance_wanted : (❰e₆❱ K M N).IsValuedIn ℤ
+instance_wanted : (❰e₆❱ K M N).IsRootSystem
+theorem_wanted e₆_hasCM_e₆ : ∀ bs : (❰e₆❱ K M N).Base, bs.HasCM (CartanMatrix.E 6)
+
+def_wanted e₇ : RootPairing (Fin 126) K M N
+instance_wanted : (❰e₇❱ K M N).IsReduced
+instance_wanted : (❰e₇❱ K M N).IsIrreducible
+instance_wanted : (❰e₇❱ K M N).IsValuedIn ℤ
+instance_wanted : (❰e₇❱ K M N).IsRootSystem
+theorem_wanted e₇_hasCM_e₇ : ∀ bs : (❰e₇❱ K M N).Base, bs.HasCM (CartanMatrix.E 7)
+
+def_wanted e₈ : RootPairing (Fin 240) K M N
+instance_wanted : (❰e₈❱ K M N).IsReduced
+instance_wanted : (❰e₈❱ K M N).IsIrreducible
+instance_wanted : (❰e₈❱ K M N).IsValuedIn ℤ
+instance_wanted : (❰e₈❱ K M N).IsRootSystem
+theorem_wanted e₈_hasCM_e₈ : ∀ bs : (❰e₈❱ K M N).Base, bs.HasCM (CartanMatrix.E 8)
+
+def_wanted f₄ : RootPairing (Fin 48) K M N
+instance_wanted : (❰f₄❱ K M N).IsReduced
+instance_wanted : (❰f₄❱ K M N).IsIrreducible
+instance_wanted : (❰f₄❱ K M N).IsValuedIn ℤ
+instance_wanted : (❰f₄❱ K M N).IsRootSystem
+theorem_wanted f₄_hasCM_f₄ : ∀ bs : (❰f₄❱ K M N).Base, bs.HasCM CartanMatrix.F₄
+
+def_wanted g₂ : RootPairing (Fin 12) K M N
+instance_wanted : (❰g₂❱ K M N).IsReduced
+instance_wanted : (❰g₂❱ K M N).IsIrreducible
+instance_wanted : (❰g₂❱ K M N).IsValuedIn ℤ
+instance_wanted : (❰g₂❱ K M N).IsRootSystem
+theorem_wanted g₂_hasCM_g₂ : ∀ bs : (❰g₂❱ K M N).Base, bs.HasCM CartanMatrix.G₂
+
+end RootPairing