Let $K$ be a field. I am interested in when there can exist a first-order definition of the set $$ \Sigma K^2 := \lbrace \sum_{i=1}^n x_i^2 \mid n \in \mathbb{N}, x_1, \ldots, x_n \in K \rbrace $$ in $K$ in the language of rings.
Clearly, if $K$ has finite Pythagoras number, then $\Sigma K^2$ has an (existential) first-order definition in $K$. In fact, for every $n \in \mathbb{N}$ there is an existential first-order formula $$ \varphi_n(x) := \exists x_1, \ldots, x_n : x = \sum_{i=1}^n x_i^2 $$ uniformly defining $\Sigma K^2$ in all fields of Pythagoras number at most $n$.
Conversely, if $K$ has infinite Pythagoras number, then there can be no first-order formula uniformly defining $\Sigma K'^2$ in all fields $K'$ elementarily equivalent to $K$. This follows from the compactness theorem.. However, this does not exclude the possibility of a field $K$ with infinite Pythagoras number and such that $\Sigma K^2$ has a first-order definition in $K$ which does not carry over to fields elementarily equivalent to $K$.
So my broad question is to understand this problem better. More specifically, I would like to understand when $\Sigma K^2$ is definable in a field $K$ of infinite Pythagoras number. Any example of a field with infinite Pythagoras number where you can prove or disprove that $\Sigma K^2$ is (existentially) definable would already be very helpful.