This is the Jordan–von Neumann theorem, and it certainly isn't simple. Unless you are another von Neumann, you are likely to need more than a few clues to find a proof of it. You can find a proof here. (The proof is given for a complex space, but there is a footnote indicating that the proof for a real space is similar.)

Edit.The J–vN theorem is actually for infinite-dimensional spaces. I suppose there might conceivably be a simpler proof for the finite-dimensional space , but I doubt it.