= Solution
Let $t^A$ be <Hermitian> color generators with $\operatorname{Tr}(t^At^B)=\delta^{AB}/2$, and take $D_\mu=\partial_\mu-ig_sG_\mu^At^A$. In explicit <spinor>, color and flavor indices the massless <kinetic term> is
$$
\mathcal L_q=i\sum_{\alpha,\beta,i,j,f}\bar q_{\alpha if}(\gamma^\mu)_{\alpha\beta}\bigl[\delta_{ij}\partial_\mu-ig_sG_\mu^A(t^A)_{ij}\bigr]q_{\beta jf}.
$$
The full <Quantum chromodynamics> <Lagrangian> also has $-G_{\mu\nu}^AG^{A\mu\nu}/4$. A local color <gauge transformation> acts as
$$
q_{\alpha if}(x)\mapsto V_{ij}(x)q_{\alpha jf}(x),\qquad \bar q_{\alpha if}\mapsto\bar q_{\alpha jf}(V^\dagger)_{ji},\qquad G_\mu\mapsto VG_\mu V^\dagger-\frac{i}{g_s}(\partial_\mu V)V^\dagger.
$$
Thus $D_\mu q\mapsto VD_\mu q$ and the <kinetic term> is <gauge-invariant>.
Since the <gauge coupling> is independent of flavor and <chirality>, the classical internal global transformations are $q_L\mapsto U_Lq_L$, $q_R\mapsto U_Rq_R$, with constant flavor matrices $U_L,U_R\in U(2)$. Equivalently these consist of $SU(2)_L\times SU(2)_R$, a common vector phase, and a singlet axial phase $q\mapsto e^{i\eta\gamma_5}q$, modulo shared discrete factors. The vector phase measures <quark> number, or <baryon number> after dividing its generator by three. The nonsinglet axial rotations may be written $q\mapsto e^{-i\theta_a\gamma_5\tau_a/2}q$. At the quantum level the singlet axial phase is anomalous; the exact continuous internal <symmetry> is $SU(2)_L\times SU(2)_R\times U(1)_V$ up to those discrete identifications. The usual spacetime Poincaré transformations also leave the theory invariant; the massless classical theory has <dilation> <symmetry>, which is broken by quantum running. These distinctions are <massless two-flavor QCD symmetries>.
At equal time the canonical relations, including all indices, are
$$
\boxed{\{q_{\alpha if}(t,\mathbf x),q_{\beta jg}^\dagger(t,\mathbf y)\}=\delta_{\alpha\beta}\delta_{ij}\delta_{fg}\delta^{(3)}(\mathbf x-\mathbf y),\qquad\{q,q\}=\{q^\dagger,q^\dagger\}=0.}
$$
No constraint removes any of the four <spinor> components of the <Dirac field> in these equal-time relations.
To evaluate the <axial charges>, first prove the elementary <bilinear> identity. For any constant <matrix> $K$ on the <spinor>/flavor space,
$$
[\int q_I^\dagger K_{IJ}q_J\,d^3y,q_K(x)]=-K_{KJ}q_J(x),\qquad [\int q^\dagger Kq\,d^3y,q_K^\dagger(x)]=q_I^\dagger(x)K_{IK}.
$$
For example $[q_I^\dagger q_J,q_K]=q_I^\dagger\{q_J,q_K\}-\{q_I^\dagger,q_K\}q_J$; the spatial delta function collapses the <integral>. This proves the charge action directly from the <canonical anticommutation relations>.
For $K_a=\gamma_5\tau_a/2$, anticommutation of $\gamma^0$ and $\gamma_5$ then gives
$$
[Q_{5,a},q]=-K_aq,\qquad[Q_{5,a},\bar q]=q^\dagger K_a\gamma^0=-\bar qK_a.
$$
The charge is an even operator, so its <commutator> obeys the ordinary product rule. Consequently
$$
[Q_{5,a},S]=-\bar q(K_a+K_a)q=-\bar q\gamma_5\tau_aq=\boxed{iP_a}.
$$
Similarly, using $\gamma_5^2=1$ and the <Pauli matrices> identity $\{\tau_a,\tau_b\}=2\delta_{ab}I$,
$$
[Q_{5,a},P_b]=-\frac i2\bar q\bigl(\tau_a\tau_b+\tau_b\tau_a\bigr)q=\boxed{-i\delta_{ab}S}.
$$
Composite fields can be defined with a common symmetry-preserving regulator; these nonsinglet identities have no color axial anomaly. This is the <nonsinglet axial charge algebra of quark densities>.
If every conserved <axial charge> left the vacuum invariant, the vacuum expectation of each such <commutator> would vanish. A nonzero <quark chiral condensate> instead makes $\langle[Q_{5,a},P_b]\rangle=-i\delta_{ab}\langle S\rangle\ne0$ for $a=b$. It therefore diagnoses spontaneous breaking of the nonsinglet <chiral symmetry>. In the massless two-flavor limit, the observed hadronic pattern is described by $SU(2)_L\times SU(2)_R\to SU(2)_V$, with three <Goldstone boson> modes identified with the <pions>. The vacuum condensate is a dynamical property of QCD, not something derivable from the canonical algebra alone. The light <pion> multiplet and absence of parity-degenerate light hadrons support this pattern; the singlet axial anomaly prevents interpreting an additional singlet mode as a fourth <Goldstone boson>.
For equal nonzero masses, add $\mathcal L_m=-m\bar qq$. It is invariant under local color <gauge transformation>s and common vector flavor rotations, but not under independent left/right rotations. The exact continuous <flavor symmetry> becomes $SU(2)_V\times U(1)_V$. The <kinetic term> and canonical <anticommutators> are unchanged, and the two proved equal-time <commutators> still hold. The charges, however, are no longer conserved:
$$
\boxed{\partial_\mu A_a^\mu=mP_a,\qquad A_a^\mu=\tfrac12\bar q\gamma^\mu\gamma_5\tau_aq,\qquad\dot Q_{5,a}=m\int P_a\,d^3x.}
$$
The first equation follows from the massive <Dirac equations>, or the last from $H_m=m\int S$ and $\dot Q=i[H,Q]$. This is the <flavor-nonsinglet axial Ward identity with equal quark masses>. The <pions> become <pseudo-Goldstone bosons> with $m_\pi^2=O(m)$, rather than exact massless modes. To leading order, in the convention $S=\bar uu+\bar dd$, $f_\pi^2m_\pi^2=-m\langle S\rangle+O(m^2)$. The condensate persists for small masses, but the <symmetry> is now explicitly broken, so the exact-symmetry Goldstone argument is replaced by an approximate one. The singlet anomaly and <gauge symmetry> are unaffected.
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