2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/129520We extend Witt's theorem to several kinds of simultaneous isometries of subspaces. We determine sufficient and necessary conditions for the extension of an isometry of subspaces $ϕ:E\to E'$ to an isometry $ϕ_V:V\to V'$ that also sends a given subspace to another, or a given self-dual flag to another, or a Witt's decomposition to another and a special self-dual flag to another. We also determine sufficient and necessary conditions for the isometry of generic flags or the simultaneous isometry of (subspace, self-dual flag) pairs.Algebraic GeometryRepresentation Theory11E39; 11E04; 15A63text