Is it true in an arbitrary field that $-1$ is a sum of two squares iff it is a sum of three squares?

The "stufe" of a field is the smallest $m$ such that $-1$ can be expressed as a sum of $m$ squares. It is a theorem of Pfister that the stufe (if it exists) is always a power of 2.

Hint $\ $ If $\rm\: -1 = a^2 + b^2 + c^2\:$ then either $\rm\:c^2 = -1\:$ or $\rm\:(a^2+b^2)/(c^2+1) = -1,\:$ which may be rewritten as a sum of two squares, since such sums $\ne 0$ form a group, using Brahmagupta for multiplication, and inversion via $\rm\ 0\ne z = x^2\! + y^2\:$ $\Rightarrow$ $\rm\:1/z = (x/z)^2 + (y/z)^2.$

Remark $\ $ The proof generalizes. Replacing the Brahmagupta composition identity by identities discovered by Pfister for quadratic forms in $\rm\:2^n\:$ variables, the proof shows that if $-1$ is a sum of $\rm\:m\:$ squares in a field, then the least value of $\rm\:m\:$ is necessarily a power of $2$, which is called the level (Stufe in German) of the field. Further, every power of $2$ is the level of some field.