We consider the class of conservative-dissipative ODE systems, which is a subclass of Lyapunov stable, linear time-invariant ODE systems. We characterize asymptotically stable, conservative-dissipative ODE systems via the hypocoercivity (theory) of their system matrices. Our main result is a concise characterization of the hypocoercivity index (an algebraic structural property of matrices with positive semi-definite Hermitian part introduced in Achleitner, Arnold, and Carlen (2018)) in terms of the short time behavior of the norm of the matrix exponential for the associated conservative-dissipative ODE system. [1] F. Achleitner, A. Arnold, and E.A. Carlen, On multi-dimensional hypocoercive BGK models, Kinet. Relat. Models 11 (2018), no. 4, 953-1009. [2] F. Achleitner, A. Arnold, and E.A. Carlen, The hypocoercivity index for the short time behavior of linear time-invariant ODE systems. Journal of Differential Equations 371 (2023), 83-115.