Dirac bracket

Remarks

Since $C$ is antisymmetric and invertible, $m$ is even. For each $c$, $\{f, \chi_c\}_D = \{f, \chi_c\} - \sum_{a, b} \{f, \chi_a\} C^{ab} C_{bc} = 0$, so the constraints commute with everything for the Dirac bracket. The Dirac bracket is antisymmetric and satisfies the Jacobi identity, and its restriction to the constraint surface $\set{\chi_a = 0}$ is the Poisson bracket of the induced symplectic structure there.

Example: $n = 2$, $\chi_1 = q_2$, $\chi_2 = p_2$. Then $C = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$ and $C^{-1} = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$. A short computation gives $$\{f, g\}_D = \frac{\partial f}{\partial q_1} \frac{\partial g}{\partial p_1} - \frac{\partial f}{\partial p_1} \frac{\partial g}{\partial q_1},$$ the bracket of the remaining pair $(q_1, p_1)$.

Non-example: for $n = 2$ and the single constraint $\chi_1 = p_2$, the matrix $C = (\{p_2, p_2\}) = (0)$ is not invertible, because $p_2$ is first class, so the Dirac bracket is not defined for it.