We study quadratic forms defined on an adelic vector space over an algebraic extension K of the rationals. Under the sole condition that a Siegel lemma holds over K, we provide height bounds for several objects naturally associated to the quadratic form, such as an isotropic subspace, a basis of isotropic vectors (when it exists) or an orthogonal basis. Our bounds involve the heights of the form and of the ambient space. In several cases, we show that the exponents of these heights are best possible. The results improve and extend previously known statements for number fields and the field of algebraic numbers.