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The ideal Chew–Goldberger–Low (CGL) plasma equations, including the double adiabatic conservation laws for the parallel ( $$p_\parallel$$ ) and perpendicular pressure ( $$p_\perp$$ ), are investigated using a Lagrangian variational principle. An Euler–Poincaré variational principle is developed and the non-canonical Poisson bracket is obtained, in which the non-canonical variables consist of the mass flux $${\boldsymbol {M}}$$ , the density $$\rho$$ , the entropy variable $$\sigma =\rho S$$ and the magnetic induction $${\boldsymbol {B}}$$ . Conservation laws of the CGL plasma equations are derived via Noether's theorem. The Galilean group leads to conservation of energy, momentum, centre of mass and angular momentum. Cross-helicity conservation arises from a fluid relabelling symmetry, and is local or non-local depending on whether the gradient of $$S$$ is perpendicular to $${\boldsymbol {B}}$$ or otherwise. The point Lie symmetries of the CGL system are shown to comprise the Galilean transformations and scalings.