In composite Higgs models the Higgs is a pseudo-Goldstone boson of a high-energy strong dynamics. We have constructed the effective chiral Lagrangian for a generic symmetric coset, restricting to CP-even bosonic operators up to four momenta which turn out to depend on seven parameters, aside from kinetic terms. Once the same sources of custodial symmetry breaking as in the Standard Model are considered, the total number of operators in the basis increases up to ten, again aside from kinetic terms. Under these assumptions, we have then particularised the discussion to three distinct frameworks: the original $SU(5)/SO(5)$ Georgi-Kaplan model, the minimal custodial-preserving $SO(5)/SO(4)$ model and the minimal $SU(3)/(SU(2)\times U(1))$ model, which intrinsically breaks custodial symmetry. The projection of the high-energy electroweak effective theory into the bosonic sector of the Standard Model is shown to match the low-energy chiral effective Lagrangian for a dynamical Higgs, and it uncovers strong relations between the operator coefficients. Finally, the relation with the bosonic basis of operators describing linear realisations of electroweak symmetry breaking is clarified.

41 pages, 2 tables version published in JHEP, typos corrected