The tangent space at a KMS state (Kubo Martin Schwinger state) can be decomposed into two subspaces in such a way that the time evolution, which is described by the modular automorphism group, satisfies the KMS condition w.r.t. each of these two subspaces. The tangent space is said to be two-typed because each tangent vector is the sum of two vectors, one in each subspace. The type of the tangent vector is conserved under time evolution. An explicit construction of the pair of subspaces is presented. The starting point is an orthonormal basis diagonalizing the modular operator. The example of the quantum spin-1/2 is worked out in detail. The constructed subspaces are of dimension 3, respectively 1. The paper is restricted to the finite-dimensional case. In this way the technicalities of handling unbounded operators are not exposed.