Problem:
An octahedron is inscribed in a cube whose edge length is 2. A cube is inscribed inside the octahedron. All the 4-fold, 3-fold and 2-fold symmetry axes of these objects coincide.
Method:
I calculated the edge length of the octahedron first. Imagine the 2D slice to look like this, then used the Pythagorean Theorem to find that the octahedron edge length was $\sqrt{2}$. All edges are equal in length for the octahedron, so I proceeded to calculate the distance from the center of any triangular face to the middle of any edge on the same face ($\frac{1}{\sqrt{6}}$). Using two different lines from center to edge on the same triangular face, I used the Pythagorean Theorem to solve for the inner cube edge ($\frac{1}{\sqrt{3}}$).
Questions:
Is my solution correct? I appreciate any suggestions or alternate solutions.
Reference Images: