What is the inradius of the octahedron with sidelength a?

1) Determine the height of one of two constituent square pyramids by considering a right triangle, using the fact that the height of the equilateral triangle is

2) Now cut the (solid) octahedron along 4 of these latter heights, i.e. across two non-adjacent vertices and two midpoints of parallel sides of the ''square base''. (Figure 1)

Consider the resultant rhombic polygon, whose sidelength is

So the question becomes: Is there a still simpler proof?