As far as I know, this is still an open problem. This is discussed in Section $5$ of the paper Finding the four squares in Lagrange's theorem by Pollack and Treviño. They mention that there is a deterministic algorithm when $n$ is a prime via quaterion multiplication, due to Bumby. Assuming a conjecture of Heath-Brown, there is a deterministic algorithm that works for all $n$. Finally, they mention that a positive proportion of all numbers can be written as the sum of four squares in deterministic polynomial time. Under the Extended Riemann Hypothesis, almost all numbers can be written as the sum of four squares in deterministic polynomial time.

As far as I know, this is still an open problem. This is discussed in Section $5$ of the paper Finding the four squares in Lagrange's theorem by Pollack and Treviño. They mention that there is a deterministic polynomial-time algorithm when $n$ is a prime via quaterion multiplication, due to Bumby. Assuming a conjecture of Heath-Brown, there is a deterministic polynomial-time algorithm that works for all $n$. Finally, they mention that a positive proportion of all numbers can be written as the sum of four squares in deterministic polynomial time. Under the Extended Riemann Hypothesis, almost all numbers can be written as the sum of four squares in deterministic polynomial time.