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The main result of this paper is a matched-pair decomposition of the space of symmetric contravariant tensors <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="fraktur">T</mi><mi mathvariant="script">Q</mi></mrow></semantics></math></inline-formula>. From this procedure two complementary Lie subalgebras of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="fraktur">T</mi><mi mathvariant="script">Q</mi></mrow></semantics></math></inline-formula> under <i>mutual</i> interaction arise. Introducing a lift operator, the matched pair decomposition of the space of Hamiltonian vector fields is determined. According to this realization, the Euler–Poincaré flows on such spaces are decomposed into two subdynamics: one is the Euler–Poincaré formulation of isentropic fluid flows, and the other one corresponds with Euler–Poincaré equations on contravariant tensors of order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>⩾</mo><mn>2</mn></mrow></semantics></math></inline-formula>.