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Final results are presented from the inclusive measurement of deep-inelasticpolarised-muon scattering on longitudinally polarised deuterons using a $^6$LiDtarget. The data were taken at $160~{\rm GeV}$ beam energy and the results areshown for the kinematic range $1~({\rm GeV}/c)^2 < Q^2 < 100~({\rm GeV}/c)^2$in photon virtuality, $0.004<x<0.7$ in the Bjorken scaling variable and $W >4~{\rm GeV}/c^2$ in the mass of the hadronic final state. The deuterondouble-spin asymmetry $A_1^{\rm d}$ and the deuteron longitudinal-spinstructure function $g_1^{\rm d}$ are presented in bins of $x$ and $Q^2$.Towards lowest accessible values of $x$, $g_1^{\rm d}$ decreases and becomesconsistent with zero within uncertainties. The presented final $g_1^{\rm d}$values together with the recently published final $g_1^{\rm p}$ values ofCOMPASS are used to again evaluate the Bjorken sum rule and perform the QCD fitto the $g_1$ world data at next-to-leading order of the strong couplingconstant. In both cases, changes in central values of the resulting numbers arewell within statistical uncertainties. The flavour-singlet axial charge $a_0$,{which is identified in the $\overline{\rm MS}$ renormalisation scheme with thetotal contribution of quark helicities to the nucleon spin}, is extracted fromonly the COMPASS deuteron data with negligible extrapolation uncertainty: $a_0(Q^2 = 3~({\rm GeV}/c)^2) = 0.32 \pm 0.02_{\rm stat} \pm0.04_{\rm syst} \pm0.05_{\rm evol}$. Together with the recent results on the proton spin structurefunction $g_1^{\rm p}$, the results on $g_1^{\rm d}$ constitute the COMPASSlegacy on the measurements of $g_1$ through inclusive spin-dependent deepinelastic scattering.

Final COMPASS results on the deuteron spin-dependent structure function $g_1^{\rm d}$ and the Bjorken sum rule