Let $${\Delta = BAG(2, q)}$$ Δ = B A G ( 2 , q ) denote the classical biaffine plane of order q, that is, the symmetric $${((q^2 - 1)_q)}$$ ( ( q 2 - 1 ) q ) configuration obtained from the classical affine plane $${\Sigma = AG(2, q)}$$ Σ = A G ( 2 , q ) of order q by omitting a point of $${\Sigma}$$ Σ together with all lines through this point. Now let $${q \geq 4}$$ q ≥ 4 be a power of a prime p and assume that $${\Delta}$$ Δ admits an embedding into the projective plane $${\Pi = PG(2, F)}$$ Π = P G ( 2 , F ) , where F is a (not necessarily commutative) field. Then this embedding extends to a projective subplane $${\Pi_0 \cong PG(2, q)}$$ Π 0 ≅ P G ( 2 , q ) of $${\Pi}$$ Π ; in particular, F has characteristic p. Consequently, $${BAG(2, q)}$$ B A G ( 2 , q ) with $${q\geq 4}$$ q ≥ 4 admits an embedding into $${PG(2, q')}$$ P G ( 2 , q ′ ) if only if q′ is a power of q. This strengthens a result of Rigby (Canad J Math 17:977–1009, 1965) in a special case while simultaneously providing a more elegant proof.