Extended Lagrange's four-square theorem

We prove the following extension of Lagrange's theorem: given a prime number p and $v_1,\dots ,v_k\in Z^4,1\le k\le 3$, such that ${\parallel v_i\parallel }^2=p$ for all $1\le i\le k$ and $〈v_i|v_j〉=0$ for all $1\le i<j\le k$, then there exists $v=(x_1,x_2,x_3,x_4)\in Z^4$ such that $〈v_i|v〉=0$ for all $1\le i\le k$ and$\parallel v\parallel =x_1^2+x_2^2+x_3^2+x_4^2=p$ This means that, in $Z^4$, any system of orthogonal vectors of norm p can be completed to a base. We conjecture that the result holds for every norm $p\ge 1$.