$k$-invariant nets over an algebraic extension of a field $k$

Let $K$ be an algebraic extension of a field $k$, let $\sigma=(\sigma_{ij})$ be an irreducible full (elementary) net of order $n\geq2$ (respectively, $n\geq3$) over $K$, while the additive subgroups $\sigma_{ij}$ are $k$-subspaces of $K$. We prove that all $\sigma_{ij}$ coincide with an intermediate subfield $P$, $k\subseteq P\subseteq K$, up to conjugation by a diagonal matrix.