C03.2 · Definitions/conventions

At a chosen basepoint of the scalar symmetric space, let \(\mathfrak p\) denote its tangent space. For a linear subspace \(\mathfrak m\subset\mathfrak p\), the Lie-triple condition is \[ [[\mathfrak m,\mathfrak m],\mathfrak m]\subset\mathfrak m. \] Let \(P\) be its orthogonal projector in the **declared tangent-space inner product**. For a basis \(X_i\) of \(\mathfrak m\), define the leakage tensor \[ \mathcal L_{ijk}=(I-P)[[X_i,X_j],X_k]. \] The zero tensor, not a dimension sum, is the required closure certificate.

If the tangent basis is not orthonormal, its Gram inverse enters the construction of the projector. Applying a Euclidean transpose to coordinates without checking that metric is not an interchangeable convention. The supplied records fix a frame, but the actual frame arrays are missing.