Before giving the definition of the decomposition, let me explain the notation used.
As shown on the left, we denote the triangle of …
For example, vertices a, ax1, and ax1x2 form slant triangle a[x1x2].
And, as shown on the right, we denote the projection of …
For example, vertices pai(a), pai(ax1), and pai(ax1x2),form flat triangle |a[x1x2]|,
where pai is the projection of “peaks and valleys” onto the hyperplane.