Consider the field of fractions $K$ of the quotient algebra $\mathbb{R}[x,y,z,t]/(x^2+y^2+z^2+t^2+1)$, where $\mathbb{R}$ is the field of real numbers and $x,y,z,t$ are variables. Clearly $-1$ is a sum of 4 squares in $K$. How can one prove that $-1$ is not a sum of 2 squares in $K$?

Serre mentions without proof this (probably known or easy) fact in a letter to Eva Bayer of May 1, 2010, and I am stuck: I cannot prove it.