surfaces of the system under study  [@sergi-theor; @sstp].

The structure of this review is as follows. In Section \[sec:qcd\], we illustrate the algebraic approach used to formulate the dynamics of a quantum subsystem embedded in a classical-like environment with canonically conjugate coordinates. In Section \[sec:csb\], we show how this formalism can be generalized to the case of a bath described by non-canonical variables, namely a collection of classical spins. Here, we will also show how an off-diagonal open-path geometric phase enters into the time evolution of the operator-valued quasi-probability function of the system. In Section \[sec:qcd-sb\], we show how the formalism allows us to also treat stochastic classical-like baths undergoing Langevin dynamics. Finally, in Section \[sec:qcnose\], we shed light on the quasi-Lie algebra established by the quantum-classical brackets and show how their antisymmetric structure is exploited to achieve thermal control of the bath DOF by means of deterministic thermostats such as the Nosé–Hoover and Nosé–Hoover chain thermostats. Our conclusions and perspectives are given in Section \[sec:cp\].

Quasi-Lie Brackets and Hybrid Quantum-Classical Systems {#sec:qcd}
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Classical and quantum dynamics share the same algebraic structure [@dirac; @balescu], which is realized by means of Poisson brackets in the classical case and commutators in 