Improving a theorem in [1] we observe that a maximal torus of an extrinsic symmetric space in a euclidean space V is itself extrinsic symmetric in some affine subspace of V. A compact extrinsic symmetric space is a submanifold X ⊂ S p−1 ⊂ R p = V such that for any point x ∈ X the reflection s x along the normal space N = N x X keeps X invariant. Every compact symmetric space X contains a maximal torus T which is unique up to congruence. If X = S n ⊂ R n+1 , the maximal torus is a great circle C

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... = X ∩ R 2 which is reflective, hence extrinsic symmetric, see [1, Theorem 4]. But for most extrinsic symmetric spaces, the maximal torus is not reflective. However, as we will show, it is an "iterated" reflective subspace, and in particular: