This section contains some explicit formulas used by FeynCalc when simpifying chains of Dirac matrices. Such relations can be found e.g. in Veltman’s Gammatrica or rederived by hand.
In the Breitenlohner-Maison-’t Hooft-Veltman scheme we are dealing with matrices in D, 4 and D-4 dimensions. Many identities of the BMHV algebra can be proven by by decomposing Dirac matrices into two pieces
\begin{align} \begin{split} \dim(\gamma^\mu) &= D, \\ \dim(\bar{\gamma}^\mu) &= 4, \\ \dim(\hat{\gamma}^\mu) &= D-4 . \end{split} \begin{split} \gamma^\mu &= \bar{\gamma}^\mu +\hat{\gamma}^\mu, \\ g^{\mu \nu} &= \bar{g}^{\mu \nu} + \hat{g}^{\mu \nu}, \\ p^\mu & = \bar{p}^\mu + \hat{p}^\mu. \end{split} \end{align}
The anticommuatators between Dirac matrices in different dimensions are given by
\begin{align} \{ \gamma^\mu, \gamma^\nu \} &= 2 g^{\mu \nu}, \\ \{ \bar{\gamma}^\mu, \bar{\gamma}^\nu \} &= \{ \gamma^\mu, \bar{\gamma}^\nu \} = 2 \bar{g}^{\mu \nu}, \\ \{ \hat{\gamma}^\mu, \hat{\gamma}^\nu \} &= \{ \gamma^\mu, \hat{\gamma}^\nu \} = 2 \hat{g}^{\mu \nu}, \\ \{ \bar{\gamma}^\mu, \hat{\gamma}^\nu \} &= 0. \end{align}
Notice that while \gamma^5 anticommutes with all other Dirac matrices in 4 dimensions, it commutes with them in D-4 dimensions. However, in D dimensions the anticommutator is not zero
\begin{align} \{ \bar{\gamma}^\mu, \gamma^5 \} &=[ \hat{\gamma}^\mu, \gamma^5 ] = 0 \\ \{ \gamma^\mu, \gamma^5 \} & = \{ \hat{\gamma}^\mu, \gamma^5 \}= 2 \hat{\gamma}^\mu \gamma^5 = 2 \gamma^5 \hat{\gamma}^\mu. \end{align}
For the chiral projectors we obtain
\begin{align} P_{L/R} \bar{\gamma}^\mu & = \bar{\gamma}^\mu P_{R/L} \\ P_{L/R} \hat{\gamma}^\mu &= \hat{\gamma}^\mu P_{L/R} \\ P_{L/R} \gamma^\mu & = \bar{\gamma}^\mu P_{R/L} + \hat{\gamma}^\mu P_{L/R} = \gamma^\mu P_{R/L} \mp \hat{\gamma}^\mu \gamma^5 \\ \gamma^\mu P_{L/R} & = P_{R/L} \bar{\gamma}^\mu + P_{L/R} \hat{\gamma}^\mu = P_{R/L} \gamma^\mu \mp \gamma^5 \hat{\gamma}^\mu \end{align}
and
\begin{align} P_{L/R} (\bar{\gamma} \cdot \bar{p} + m) &= \bar{\gamma} \cdot \bar{p} P_{R/L} + m P_{L/R} = (\bar{\gamma} \cdot \bar{p} - m) P_{R/L} + m \\ P_{L/R} (\bar{\gamma} \cdot \hat{p} + m) & = (\bar{\gamma} \cdot \hat{p} + m) P_{L/R} \\ P_{L/R} (\bar{\gamma} \cdot p + m) &= (\bar{\gamma} \cdot \hat{p} + m) P_{L/R} + \bar{\gamma} \cdot \bar{p} P_{R/L} = (\bar{\gamma} \cdot \bar{p} - m) P_{R/L} + \bar{\gamma} \cdot \hat{p} P_{L/R} + m \\ \nonumber \\ \nonumber %%%%%%%%%%%%%% (\bar{\gamma} \cdot \bar{p} + m) P_{L/R} &= P_{R/L} \bar{\gamma} \cdot \bar{p} + P_{L/R} m = P_{R/L} (\bar{\gamma} \cdot \bar{p} - m) + m \\ (\bar{\gamma} \cdot \hat{p} + m) P_{L/R} & = P_{L/R} (\bar{\gamma} \cdot \hat{p} + m) \\ (\bar{\gamma} \cdot p + m) P_{L/R} &= P_{L/R} ( \bar{\gamma} \cdot \hat{p} + m) + P_{R/L} \bar{\gamma} \cdot \bar{p} = P_{R/L} (\bar{\gamma} \cdot \bar{p} - m) + P_{L/R} \bar{\gamma} \cdot \hat{p} + m \end{align}
Notice that package TRACER resolves the redundancy of
having \gamma^\mu = \bar{\gamma}^\mu +
\hat{\gamma}^\mu by eliminating \bar{\gamma}^\mu and offering a function that
reintroduces it at the end of the calculation.
Contractions of Dirac matrices and vectors with the metric read
\begin{align} \begin{split} g^{\mu \nu} \gamma_\nu &= \gamma^\mu, \\ \bar{g}^{\mu \nu} \bar{\gamma}_\nu &= g^{\mu \nu} \bar{\gamma}_\nu = \bar{g}^{\mu \nu} \gamma_\nu = \bar{\gamma}^\mu, \\ \hat{g}^{\mu \nu} \hat{\gamma}_\nu &= g^{\mu \nu} \hat{\gamma}_\nu = \hat{g}^{\mu \nu} \gamma_\nu = \hat{\gamma}^\mu, \\ \bar{g}^{\mu \nu} \hat{\gamma}_\nu &= \hat{g}^{\mu \nu} \bar{\gamma}_\nu = 0, \end{split} \begin{split} g^{\mu \nu} p_\nu &= p^\mu, \\ \bar{g}^{\mu \nu} \bar{p}_\nu &= g^{\mu \nu} \bar{p}_\nu = \bar{g}^{\mu \nu} p_\nu = \bar{p}^\mu, \\ \hat{g}^{\mu \nu} \hat{p}_\nu &= g^{\mu \nu} \hat{p}_\nu = \hat{g}^{\mu \nu} p_\nu = \hat{p}^\mu, \\ \bar{g}^{\mu \nu} \hat{p}_\nu &= \hat{g}^{\mu \nu} \bar{p}_\nu = 0. \end{split} \end{align}
Contractions of the metric with itself
\begin{align} \begin{split} g^{\mu \nu} g_{\nu \rho} & = g^\mu_\rho \\ \bar{g}^{\mu \nu} \bar{g}_{\nu \rho} & =g^{\mu \nu} \bar{g}_{\nu \rho} = \bar{g}^{\mu \nu} g_{\nu \rho} = \bar{g}^\mu_\rho \\ \hat{g}^{\mu \nu} \hat{g}_{\nu \rho} & = g^{\mu \nu} \hat{g}_{\nu \rho} = \hat{g}^{\mu \nu} g_{\nu \rho}= \hat{g}^\mu_\rho \\ \bar{g}^{\mu \nu} \hat{g}_{\nu \rho} &= \hat{g}^{\mu \nu} \bar{g}_{\nu \rho} = 0, \end{split} \begin{split} g^{\mu \nu} g_{\mu \nu} & = d, \\ \bar{g}^{\mu \nu} \bar{g}_{\mu \nu} & = g^{\mu \nu} \bar{g}_{\mu \nu} = \bar{g}^{\mu \nu} g_{\mu \nu} = 4, \\ \hat{g}^{\mu \nu} \hat{g}_{\mu \nu} & = g^{\mu \nu} \hat{g}_{\mu \nu} = \hat{g}^{\mu \nu} g_{\mu \nu} = d-4, \\ \bar{g}^{\mu \nu} \hat{g}_{\mu \nu} &= \hat{g}^{\mu \nu} \bar{g}_{\mu \nu} = 0. \end{split} \end{align}
Contractions of Dirac matrices and vectors with themselves
\begin{align} \begin{split} \gamma^\mu \gamma_\mu &= D, \\ \bar{\gamma}^\mu \bar{\gamma}_\mu &= \gamma^\mu \bar{\gamma}_\mu = \bar{\gamma}^\mu \gamma_\mu = 4, \\ \hat{\gamma}^\mu \hat{\gamma}_\mu &= \gamma^\mu \hat{\gamma}_\mu = \hat{\gamma}^\mu \gamma_\mu = D-4, \\ \bar{\gamma}^\mu \hat{\gamma}_\mu &= \hat{\gamma}^\mu \bar{\gamma}_\mu = 0, \end{split} \begin{split} p^\mu p_\mu &= p^2, \\ \bar{p}^\mu \bar{p}_\mu &= \bar{p}^\mu p_\mu = p^\mu \bar{p}_\mu = \bar{p}^2, \\ \hat{p}^\mu \hat{p}_\mu &= \hat{p}^\mu p_\mu = p^\mu \hat{p}_\mu = \hat{p}^2, \\ \bar{p}^\mu \hat{p}_\mu &= \hat{p}^\mu \bar{p}_\mu = 0. \end{split} \end{align}
Dirac slashes
\begin{align} \begin{split} \gamma^\mu p_\mu &= \gamma \cdot p, \\ \bar{\gamma}^\mu \bar{p}_\mu &= \bar{\gamma}^\mu p_\mu = \gamma^\mu \bar{p}_\mu = \bar{\gamma} \cdot \bar{p}, \\ \hat{\gamma}^\mu p_\mu &= \hat{\gamma}^\mu p_\mu = \gamma^\mu \hat{p}_\mu = \hat{\gamma} \cdot \hat{p}, \\ \bar{\gamma}^\mu \hat{p}_\mu &= \hat{\gamma}^\mu \bar{p}_\mu = 0. \\ \end{split} \end{align}
Index pairs with one, two, three, four or five free indices
\begin{align} \begin{split} \gamma^\mu \gamma^\nu \gamma_\mu &= -(d-2) \gamma^\nu, \\ \gamma^\mu \bar{\gamma}^\nu \gamma_\mu &= -(d-2) \bar{\gamma}^\nu, \\ \gamma^\mu \hat{\gamma}^\nu \gamma_\mu &= -(d-2) \hat{\gamma}^\nu, \end{split} \quad \quad \begin{split} \bar{\gamma}^\mu \bar{\gamma}^\nu \bar{\gamma}_\mu &= -2 \bar{\gamma}^\nu, \\ \bar{\gamma}^\mu \gamma^\nu \bar{\gamma}_\mu &= -4 \gamma^\nu +2 \bar{\gamma}^\nu, \\ \bar{\gamma}^\mu \hat{\gamma}^\nu \bar{\gamma}_\mu &= -4 \hat{\gamma}^\nu, \\ \end{split} \quad \quad \quad \quad \begin{split} \hat{\gamma}^\mu \hat{\gamma}^\nu \hat{\gamma}_\mu &= -(d-6) \hat{\gamma}^\nu, \\ \hat{\gamma}^\mu \gamma^\nu \hat{\gamma}_\mu &= -(d-4) \gamma^\nu + 2\hat{\gamma}^\nu, \\ \hat{\gamma}^\mu \bar{\gamma}^\nu \hat{\gamma}_\mu &= -(d-4) \bar{\gamma}^\nu, \end{split} \end{align}
\begin{align} \gamma^\mu \gamma^\nu \gamma^\rho \gamma_\mu &= (d-4) \gamma^\nu \gamma^\rho + 4 g^{\nu \rho} I, \\ \gamma^\mu \bar{\gamma}^\nu \bar{\gamma}^\rho \gamma_\mu &= (d-4) \bar{\gamma}^\nu \bar{\gamma}^\rho + 4 \bar{g}^{\nu \rho} I, \\ \gamma^\mu \hat{\gamma}^\nu \hat{\gamma}^\rho\gamma_\mu &= (d-4) \hat{\gamma}^\nu \hat{\gamma}^\rho + 4 \hat{g}^{\nu \rho} I, \\ \bar{\gamma}^\mu \bar{\gamma}^\nu \bar{\gamma}^\rho \bar{\gamma}_\mu &= 4 \bar{g}^{\nu \rho}, \\ \bar{\gamma}^\mu \gamma^\nu \gamma^\rho \bar{\gamma}_\mu &= 4 \gamma^\nu \gamma^\rho - 2 \bar{\gamma}^\nu \gamma^\rho + 2\bar{\gamma}^\rho \gamma^\nu, \\ \bar{\gamma}^\mu \hat{\gamma}^\nu \hat{\gamma}^\rho \bar{\gamma}_\mu &= 4 \hat{\gamma}^\nu \hat{\gamma}^\rho, \\ \hat{\gamma}^\mu \hat{\gamma}^\nu \hat{\gamma}^\rho \hat{\gamma}_\mu &= (d-8) \hat{\gamma}^\nu \hat{\gamma}^\rho + 4 \hat{g}^{\nu \rho} I, \\ \hat{\gamma}^\mu \gamma^\nu \gamma^\rho \hat{\gamma}_\mu &= (d-4) \gamma^\nu \gamma^\rho - 2 \hat{\gamma}^\nu \gamma^\rho + 2\hat{\gamma}^\rho \gamma^\nu, \\ \hat{\gamma}^\mu \bar{\gamma}^\nu \bar{\gamma}^\rho \hat{\gamma}_\mu &= (d-4) \bar{\gamma}^\nu \bar{\gamma}^\rho, \end{align}
\begin{align} \gamma^\mu \gamma^\nu \gamma^\rho \gamma^\sigma \gamma_\mu &= -(d-4) \gamma^\nu \gamma^\rho \gamma^\sigma -2 \gamma^\sigma \gamma^\rho \gamma^\nu \\ \gamma^\mu \bar{\gamma}^\nu \bar{\gamma}^\rho \bar{\gamma}^\sigma \gamma_\mu &=-(d-4) \bar{\gamma}^\nu \bar{\gamma}^\rho \bar{\gamma}^\sigma -2 \bar{\gamma}^\sigma \bar{\gamma}^\rho \bar{\gamma}^\nu \\ \gamma^\mu \hat{\gamma}^\nu \hat{\gamma}^\rho \hat{\gamma}^\sigma \gamma_\mu &= -(d-4) \hat{\gamma}^\nu \hat{\gamma}^\rho \hat{\gamma}^\sigma -2 \hat{\gamma}^\sigma \hat{\gamma}^\rho \hat{\gamma}^\nu \\ \bar{\gamma}^\mu \bar{\gamma}^\nu \bar{\gamma}^\rho \bar{\gamma}^\sigma \bar{\gamma}_\mu &= -2 \bar{\gamma}^\sigma \bar{\gamma}^\rho \bar{\gamma}^\nu \\ \bar{\gamma}^\mu \gamma^\nu \gamma^\rho \gamma^\sigma \bar{\gamma}_\mu &= -4 \gamma^\nu \gamma^\rho \gamma^\sigma + 2 \bar{\gamma}^\nu \gamma^\rho \gamma^\sigma - 2\bar{\gamma}^\rho \gamma^\nu \gamma^\sigma + 2 \bar{\gamma}^\sigma \gamma^\nu \gamma^\rho, \\ \bar{\gamma}^\mu \hat{\gamma}^\nu \hat{\gamma}^\rho \hat{\gamma}^\sigma \bar{\gamma}_\mu &= - 4 \hat{\gamma}^\nu \hat{\gamma}^\rho \hat{\gamma}^\sigma , \\ \hat{\gamma}^\mu \hat{\gamma}^\nu \hat{\gamma}^\rho \hat{\gamma}^\sigma \hat{\gamma}_\mu &= -(d-8) \hat{\gamma}^\nu \hat{\gamma}^\rho \hat{\gamma}^\sigma -2 \hat{\gamma}^\sigma \hat{\gamma}^\rho \hat{\gamma}^\nu \\ \hat{\gamma}^\mu \gamma^\nu \gamma^\rho \gamma^\sigma \hat{\gamma}_\mu &= -(d-4) \gamma^\nu \gamma^\rho \gamma^\sigma + 2 \hat{\gamma}^\nu \gamma^\rho \gamma^\sigma - 2\hat{\gamma}^\rho \gamma^\nu \gamma^\sigma + 2 \hat{\gamma}^\sigma \gamma^\nu \gamma^\rho, \\ \hat{\gamma}^\mu \bar{\gamma}^\nu \bar{\gamma}^\rho \bar{\gamma}^\sigma \hat{\gamma}_\mu &= -(d-4) \bar{\gamma}^\nu \bar{\gamma}^\rho \bar{\gamma}^\sigma, \end{align}
\begin{align} \gamma^\mu \gamma^\nu \gamma^\rho \gamma^\sigma \gamma^\tau \gamma_\mu &= (d-4) \gamma^\nu \gamma^\rho \gamma^\sigma \gamma^\tau + 2 \gamma^\sigma \gamma^\rho \gamma^\nu \gamma^\tau + 2 \gamma^\tau \gamma^\nu \gamma^\rho \gamma^\sigma \\ \gamma^\mu \bar{\gamma}^\nu \bar{\gamma}^\rho \bar{\gamma}^\sigma \bar{\gamma}^\tau \gamma_\mu &= (d-4) \bar{\gamma}^\nu \bar{\gamma}^\rho \bar{\gamma}^\sigma \bar{\gamma}^\tau + 2 \bar{\gamma}^\sigma \bar{\gamma}^\rho \bar{\gamma}^\nu \bar{\gamma}^\tau + 2 \bar{\gamma}^\tau \bar{\gamma}^\nu \bar{\gamma}^\rho \bar{\gamma}^\sigma \\ \gamma^\mu \hat{\gamma}^\nu \hat{\gamma}^\rho \hat{\gamma}^\sigma \hat{\gamma}^\tau \gamma_\mu &= (d-4) \hat{\gamma}^\nu \hat{\gamma}^\rho \hat{\gamma}^\sigma \hat{\gamma}^\tau +2 \hat{\gamma}^\sigma \hat{\gamma}^\rho \hat{\gamma}^\nu \hat{\gamma}^\tau + 2\hat{\gamma}^\tau \hat{\gamma}^\nu \hat{\gamma}^\rho \hat{\gamma}^\sigma \\ \bar{\gamma}^\mu \bar{\gamma}^\nu \bar{\gamma}^\rho \bar{\gamma}^\sigma \bar{\gamma}^\tau \bar{\gamma}_\mu &= 2 \bar{\gamma}^\sigma \bar{\gamma}^\rho \bar{\gamma}^\nu \bar{\gamma}^\tau + 2 \bar{\gamma}^\tau \bar{\gamma}^\nu \bar{\gamma}^\rho \bar{\gamma}^\sigma \\ \bar{\gamma}^\mu \gamma^\nu \gamma^\rho \gamma^\sigma \gamma^\tau \bar{\gamma}_\mu &= 4 \gamma^\nu \gamma^\rho \gamma^\sigma \gamma^\tau \\ &- 2 \bar{\gamma}^\nu \gamma^\rho \gamma^\sigma \gamma^\tau + 2\bar{\gamma}^\rho \gamma^\nu \gamma^\sigma \gamma^\tau - 2 \bar{\gamma}^\sigma \gamma^\nu \gamma^\rho \gamma^\tau + 2 \bar{\gamma}^\tau \gamma^\nu \gamma^\rho \gamma^\sigma, \\ \bar{\gamma}^\mu \hat{\gamma}^\nu \hat{\gamma}^\rho \hat{\gamma}^\sigma \hat{\gamma}^\tau \bar{\gamma}_\mu &= 4 \hat{\gamma}^\nu \hat{\gamma}^\rho \hat{\gamma}^\sigma \hat{\gamma}^\tau , \\ \hat{\gamma}^\mu \hat{\gamma}^\nu \hat{\gamma}^\rho \hat{\gamma}^\sigma \hat{\gamma}^\tau \hat{\gamma}_\mu &= (d-8) \hat{\gamma}^\nu \hat{\gamma}^\rho \hat{\gamma}^\sigma \hat{\gamma}^\tau + 2 \hat{\gamma}^\sigma \hat{\gamma}^\rho \hat{\gamma}^\nu \hat{\gamma}^\tau + 2 \hat{\gamma}^\tau \hat{\gamma}^\nu \hat{\gamma}^\rho \hat{\gamma}^\sigma \\ \hat{\gamma}^\mu \gamma^\nu \gamma^\rho \gamma^\sigma \gamma^\tau \hat{\gamma}_\mu &= (d-4) \gamma^\nu \gamma^\rho \gamma^\sigma \gamma^\tau \\ &- 2 \hat{\gamma}^\nu \gamma^\rho \gamma^\sigma \gamma^\tau + 2\hat{\gamma}^\rho \gamma^\nu \gamma^\sigma \gamma^\tau - 2 \hat{\gamma}^\sigma \gamma^\nu \gamma^\rho \gamma^\tau + 2 \hat{\gamma}^\tau \gamma^\nu \gamma^\rho \gamma^\sigma, \\ \hat{\gamma}^\mu \bar{\gamma}^\nu \bar{\gamma}^\rho \bar{\gamma}^\sigma \bar{\gamma}^\tau \hat{\gamma}_\mu &= (d-4) \bar{\gamma}^\nu \bar{\gamma}^\rho \bar{\gamma}^\sigma \bar{\gamma}^\tau, \end{align}
\begin{align} \gamma^\mu \gamma^\nu \gamma^\rho \gamma^\sigma \gamma^\tau \gamma^\kappa \gamma_\mu =& -(d-4) \gamma^\nu \gamma^\rho \gamma^\sigma \gamma^\tau \gamma^\kappa \\ &- 2 \gamma^\sigma \gamma^\rho \gamma^\nu \gamma^\tau \gamma^\kappa - 2 \gamma^\tau \gamma^\nu \gamma^\rho \gamma^\sigma \gamma^\kappa + 2\gamma^\kappa \gamma^\nu \gamma^\rho \gamma^\sigma \gamma^\tau \\ \gamma^\mu \bar{\gamma}^\nu \bar{\gamma}^\rho \bar{\gamma}^\sigma \bar{\gamma}^\tau \bar{\gamma}^\kappa \gamma_\mu =& -(d-4) \bar{\gamma}^\nu \bar{\gamma}^\rho \bar{\gamma}^\sigma \bar{\gamma}^\tau \bar{\gamma}^\kappa \\ & - 2 \bar{\gamma}^\sigma \bar{\gamma}^\rho \bar{\gamma}^\nu \bar{\gamma}^\tau \bar{\gamma}^\kappa - 2 \bar{\gamma}^\tau \bar{\gamma}^\nu \bar{\gamma}^\rho \bar{\gamma}^\sigma \bar{\gamma}^\kappa + 2\bar{\gamma}^\kappa \bar{\gamma}^\nu \bar{\gamma}^\rho \bar{\gamma}^\sigma \bar{\gamma}^\tau\\ \gamma^\mu \hat{\gamma}^\nu \hat{\gamma}^\rho \hat{\gamma}^\sigma \hat{\gamma}^\tau \hat{\gamma}^\kappa \gamma_\mu =& -(d-4) \hat{\gamma}^\nu \hat{\gamma}^\rho \hat{\gamma}^\sigma \hat{\gamma}^\tau \hat{\gamma}^\kappa \\ & - 2 \hat{\gamma}^\sigma \hat{\gamma}^\rho \hat{\gamma}^\nu \hat{\gamma}^\tau \hat{\gamma}^\kappa - 2 \hat{\gamma}^\tau \hat{\gamma}^\nu \hat{\gamma}^\rho \hat{\gamma}^\sigma \hat{\gamma}^\kappa + 2\hat{\gamma}^\kappa \hat{\gamma}^\nu \hat{\gamma}^\rho \hat{\gamma}^\sigma \hat{\gamma}^\tau \\ \bar{\gamma}^\mu \bar{\gamma}^\nu \bar{\gamma}^\rho \bar{\gamma}^\sigma \bar{\gamma}^\tau \bar{\gamma}^\kappa \bar{\gamma}_\mu & = 2 \bar{\gamma}^\tau \bar{\gamma}^\sigma \bar{\gamma}^\rho \bar{\gamma}^\nu \bar{\gamma}^\kappa + 2 \bar{\gamma}^\kappa \bar{\gamma}^\nu \bar{\gamma}^\rho \bar{\gamma}^\sigma \bar{\gamma}^\tau = - 2 \bar{\gamma}^\kappa \bar{\gamma}^\tau \bar{\gamma}^\sigma \bar{\gamma}^\rho \bar{\gamma}^\nu \\ \bar{\gamma}^\mu \gamma^\nu \gamma^\rho \gamma^\sigma \gamma^\tau \gamma^\kappa \bar{\gamma}_\mu =& -4 \gamma^\nu \gamma^\rho \gamma^\sigma \gamma^\tau \gamma^\kappa + 2 \bar{\gamma}^\nu \gamma^\rho \gamma^\sigma \gamma^\tau \gamma^\kappa - 2\bar{\gamma}^\rho \gamma^\nu \gamma^\sigma \gamma^\tau \gamma^\kappa \\ &+ 2 \bar{\gamma}^\sigma \gamma^\nu \gamma^\rho \gamma^\tau \gamma^\kappa - 2 \bar{\gamma}^\tau \gamma^\nu \gamma^\rho \gamma^\sigma \gamma^\kappa + 2\bar{\gamma}^\kappa \gamma^\nu \gamma^\rho \gamma^\sigma \gamma^\tau \\ \bar{\gamma}^\mu \hat{\gamma}^\nu \hat{\gamma}^\rho \hat{\gamma}^\sigma \hat{\gamma}^\tau \hat{\gamma}^\kappa \bar{\gamma}_\mu =& -4 \hat{\gamma}^\nu \hat{\gamma}^\rho \hat{\gamma}^\sigma \hat{\gamma}^\tau \hat{\gamma}^\kappa , \\ \hat{\gamma}^\mu \hat{\gamma}^\nu \hat{\gamma}^\rho \hat{\gamma}^\sigma \hat{\gamma}^\tau \hat{\gamma}^\kappa \hat{\gamma}_\mu =& -(d-8) \hat{\gamma}^\nu \hat{\gamma}^\rho \hat{\gamma}^\sigma \hat{\gamma}^\tau \hat{\gamma}^\kappa \\ &- 2 \hat{\gamma}^\sigma \hat{\gamma}^\rho \hat{\gamma}^\nu \hat{\gamma}^\tau \hat{\gamma}^\kappa - 2 \hat{\gamma}^\tau \hat{\gamma}^\nu \hat{\gamma}^\rho \hat{\gamma}^\sigma \hat{\gamma}^\kappa + 2 \hat{\gamma}^\kappa \hat{\gamma}^\nu \hat{\gamma}^\rho \hat{\gamma}^\sigma \hat{\gamma}^\tau \\ \hat{\gamma}^\mu \gamma^\nu \gamma^\rho \gamma^\sigma \gamma^\tau \gamma^\kappa \hat{\gamma}_\mu =& -(d-4) \gamma^\nu \gamma^\rho \gamma^\sigma \gamma^\tau \gamma^\kappa + 2 \hat{\gamma}^\nu \gamma^\rho \gamma^\sigma \gamma^\tau \gamma^\kappa - 2\hat{\gamma}^\rho \gamma^\nu \gamma^\sigma \gamma^\tau \gamma^\kappa \\ & + 2 \hat{\gamma}^\sigma \gamma^\nu \gamma^\rho \gamma^\tau \gamma^\kappa - 2 \hat{\gamma}^\tau \gamma^\nu \gamma^\rho \gamma^\sigma \gamma^\kappa +2 \hat{\gamma}^\kappa \gamma^\nu \gamma^\rho \gamma^\sigma \gamma^\tau, \\ \hat{\gamma}^\mu \bar{\gamma}^\nu \bar{\gamma}^\rho \bar{\gamma}^\sigma \bar{\gamma}^\tau \bar{\gamma}^\kappa \hat{\gamma}_\mu =& -(d-4) \bar{\gamma}^\nu \bar{\gamma}^\rho \bar{\gamma}^\sigma \bar{\gamma}^\tau \bar{\gamma}^\kappa, \end{align}
If the first and the last matrix are in different dimensions, we can always write them as
\begin{align} \gamma^\mu \dots \bar{\gamma}_\mu & = \bar{\gamma}^\mu \dots \gamma_\mu = \bar{\gamma}^\mu \dots \bar{\gamma}_\mu, \\ \gamma^\mu \dots \hat{\gamma}_\mu & = \hat{\gamma}^\mu \dots \gamma_\mu = \hat{\gamma}^\mu \dots \hat{\gamma}_\mu, \\ \bar{\gamma}^\mu \dots \hat{\gamma}_\mu & = \hat{\gamma}^\mu \dots \bar{\gamma}_\mu = 0. \end{align}
For general index pairs we have that
\begin{align} \gamma^\mu \bar{\gamma}^{\nu_1} \dots \bar{\gamma}^{\nu_n} \gamma_\mu &= \bar{\gamma}^\mu \bar{\gamma}^{\nu_1} \dots \bar{\gamma}^{\nu_n} \bar{\gamma}_\mu + (-1)^n (D-4) \bar{\gamma}^{\nu_1} \dots \bar{\gamma}^{\nu_n}, \\ \gamma^\mu \hat{\gamma}^{\nu_1} \dots \hat{\gamma}^{\nu_n} \gamma_\mu &= \hat{\gamma}^\mu \hat{\gamma}^{\nu_1} \dots \hat{\gamma}^{\nu_n} \hat{\gamma}_\mu + 4 (-1)^n \hat{\gamma}^{\nu_1} \dots \hat{\gamma}^{\nu_n}, \\ \bar{\gamma}^\mu \hat{\gamma}^{\nu_1} \dots \hat{\gamma}^{\nu_n} \bar{\gamma}_\mu &= 4 (-1)^n \, \hat{\gamma}^{\nu_1} \dots \hat{\gamma}^{\nu_n}, \\ \hat{\gamma}^\mu \bar{\gamma}^{\nu_1} \dots \bar{\gamma}^{\nu_n} \hat{\gamma}_\mu &= (D-4) (-1)^n \, \bar{\gamma}^{\nu_1} \dots \bar{\gamma}^{\nu_n}, \end{align}
This means that if we have a general formula for \gamma^\mu \gamma^{\nu_1} \dots \gamma^{\nu_n} \gamma_\mu in D dimensions, we can easily obtain 7 of 9 possible combinations of dimensions. The other two cases are special and related with each other
\begin{align} \bar{\gamma}^\mu \gamma^{\nu_1} \dots \gamma^{\nu_n} \bar{\gamma}_\mu = \gamma^\mu \gamma^{\nu_1} \dots \gamma^{\nu_n} \gamma_\mu - \hat{\gamma}^\mu \gamma^{\nu_1} \dots \gamma^{\nu_n} \hat{\gamma}_\mu. \end{align}
If we know \bar{\gamma}^\mu \gamma^{\nu_1} \dots \gamma^{\nu_n} \bar{\gamma}_\mu we can easily compute \hat{\gamma}^\mu \gamma^{\nu_1} \dots \gamma^{\nu_n} \hat{\gamma}_\mu and vice versa.
For purely four dimensional chains it is known that if the number of the Dirac matrices between the index pair is odd, then
\begin{align} \bar{\gamma}^\mu \bar{\gamma}^{\nu_1} \dots \bar{\gamma}^{\nu_{2n+1}} \bar{\gamma}_\mu & = -2 \bar{\gamma}^{\nu_{2n+1}} \dots \bar{\gamma}^{\nu_1}. \end{align}
This can be trivially generalized to even chains, i.e.
\begin{align} \bar{\gamma}^\mu \bar{\gamma}^{\nu_1} \dots \bar{\gamma}^{\nu_{2n+1}} \bar{\gamma}^\rho \bar{\gamma}_\mu & = 2 \bar{\gamma}^{\nu_{2n+1}} \dots \bar{\gamma}^{\nu_1} \bar{\gamma}^\rho + 2 \bar{\gamma}^\rho \bar{\gamma}^{\nu_1} \dots \bar{\gamma}^{\nu_{2n+1}} \end{align}
Following formula for \gamma^\mu \gamma^{\nu_1} \dots \gamma^{\nu_n} \gamma_\mu in d dimensions and n\geq 3 is given in Veltman’s Gammatrica
\begin{align} \gamma^\mu \gamma^{\nu_1} \dots \gamma^{\nu_n} \gamma_\mu &= (d-4)(-1)^n \gamma^{\nu_1} \dots \gamma^{\nu_n} + 2(-1)^n \gamma^{\nu_3} \gamma^{\nu_2} \gamma^{\nu_1} \gamma^{\nu_4}\dots \gamma^{\nu_n} \\ & + 2 \sum_{j=4}^m (-1)^{n-j} \gamma^{\nu_j} \gamma^{\nu_1} \dots \gamma^{\nu_{j-1}} \gamma^{\nu_{j+1}} \dots \gamma^{\nu_{n}}. \end{align}
Another useful and more compact formula for the same expression with n \geq 2 was derived my R. Mertig
\begin{align} \gamma^\mu \gamma^{\nu_1} \dots \gamma^{\nu_n} \gamma_\mu &= (-1)^n \biggl \{ (d-2n) \gamma^{\nu_1} \dots \gamma^{\nu_n} \\ & -4 \sum_{i=1}^{l-1} \sum_{j=i+1}^{l} (-1)^{j-i} \gamma^{\nu_1} \dots \gamma^{\nu_{i-1}} \gamma^{\nu_{i+1}} \dots \gamma^{\nu_{j-1}} \gamma^{\nu_{j+1}} \dots \gamma^{\nu_n} g_{{\mu_i} {\mu_j}} \biggr \} \end{align}