Sum of three squares of polynomials

Question

In the polynomial ring $\mathbb{R}[x,y,z,u,v,w]$ with $6$ variables, I would like to know if there exist $f,g,h\in \mathbb{R}[x,y,z,u,v,w]$ such that $$ (x^2+y^2+z^2)(u^2+v^2+w^2)=f^2+g^2+h^2. $$ Any reference/proof about this problem?

Thanks.