 The restriction of "q" to "H" has defect line "L", so there is an induced quadratic form "Q" on the 4-dimensional "H"/"L" that one checks is non-degenerate and non-split.The zero scheme of "Q" in "H"/"L" defines a smooth quadric surface "X" in the associated projective 3-space over "k".Over an algebraic closure of "k", "X" is a product of two projective lines, so by a descent argument "X" is the Weil restriction to "k" of the projective line over a quadratic étale algebra "K".