The following relations can be written for the generators $\left(γ^μ: 0 ≤ μ < n\right)$ of a Clifford algebra, over some $n$ dimensional vector space, with a basis $\left(θ^μ: 0 ≤ μ < n\right)$ equipped with a metric $\left(g^{μν}: 0 ≤ μ,ν < n\right)$ (where $g^{μν} = g^{νμ}$):
$$
γ^μ γ^ν = g^{μν} + γ^{μν},\\
γ^μ γ^{νρ} = g^{μν} γ^ρ - g^{μρ} γ^ν + γ^{μνρ},\\
γ^μ γ^{νρσ} = g^{μν} γ^{ρσ} - g^{μρ} γ^{νσ} + g^{μσ} γ^{νρ} + γ^{μνρσ},\\
⋯
$$
where
$$
γ^{μν} = \frac{γ^μ γ^ν - γ^ν γ^μ}{2},\\
γ^{μνρ} = \frac{γ^μ γ^{νρ} - γ^ν γ^{μρ} + γ^ρ γ^{μν}}{3},\\
γ^{μνρσ} = \frac{γ^μ γ^{νρσ} - γ^ν γ^{μρσ} + γ^ρ γ^{μνσ} - γ^σ γ^{μνρ}}{4},\\
⋯
$$
are each completely anti-symmetric in their indices and are anti-symmetric sums over their respective permutations; e.g. $γ^{μνρ}$ is summed over permutations of $γ^μ$, $γ^ν$ and $γ^ρ$, with $±$ sign depending on the parity of the permutation, it's just easier and more compact to write it the way I wrote it above.
In fact, you can think of the Clifford algebra as being an algebra over all the exterior algebra associated with the underlying space, such that
$$
γ\left(A + A_μ θ^μ + \frac{A_{μν} θ^μ ∧ θ^ν}{2!} + \frac{A_{μνρ} θ^μ ∧ θ^ν ∧ θ^ρ}{3!} + ⋯\right) = A + A_μ γ^μ + \frac{A_{μν} γ^{μν}}{2!} + \frac{A_{μνρ} γ^{μνρ}}{3!} + ⋯
$$
(using the Summation Convention on repeated indices). Then the rule for the product is that if $v$ is a vector and $α$ is a multi-vector, then
$$γ(v) γ(α) = γ(v · α) + γ(v ∧ α),$$
where the contraction on the left is defined, for scalars $u$, vectors $v = v_μ θ^μ$ and $w = w_ν θ^ν$ and multi-vectors $α$, recursively by
$$v · u = 0,\quad v · (w ∧ α) = g(v,w) α - w ∧ (v · α)\quad (g(v,w) = g^{μν} v_μ w_ν = v · w).$$
You could, just as well, treat it as an expansion of the exterior algebra, itself, drop all of the $γ$'s and just write:
$$γ^μ = θ^μ,\quad v α = v · α + v ∧ α,$$
for vectors $v$ and multi-vectors $α$.
With all of that being said, higher-order products can be written as follows:
$$\begin{align}
γ^μ γ^ν γ^{ρσ}
&= γ^μ \left(g^{νρ} γ^σ - g^{νσ} γ^ρ + γ^{νρσ}\right)\\
&= g^{νρ} γ^μ γ^σ - g^{νσ} γ^μ γ^ρ + γ^μ γ^{νρσ}\\
&= g^{νρ} \left(g^{μσ} + γ^{μσ}\right) - g^{νσ} \left(g^{μρ} + γ^{μρ}\right)
+ \left(g^{μν} γ^{ρσ} - g^{μρ} γ^{νσ} + g^{μσ} γ^{νρ} + γ^{μνρσ}\right)\\
&= \left(g^{νρ} g^{μσ} - g^{νσ} g^{μρ}\right) + \left(g^{νρ} γ^{μσ} - g^{νσ} γ^{μρ} + g^{μν} γ^{ρσ} - g^{μρ} γ^{νσ} + g^{μσ} γ^{νρ}\right) + γ^{μνρσ}.
\end{align}$$
Thus, anti-symmetrizing over $μ$ and $ν$, we get:
$$\begin{align}
γ^{μν} γ^{ρσ}
&= \frac{γ^μ γ^ν - γ^ν γ^μ}2 γ^{ρσ}\\
&= \left(g^{νρ} g^{μσ} - g^{νσ} g^{μρ}\right) + \left(g^{νρ} γ^{μσ} - g^{νσ} γ^{μρ} - g^{μρ} γ^{νσ} + g^{μσ} γ^{νρ}\right) + γ^{μνρσ}.
\end{align}$$
Swapping indices ($μ ↔ ρ$ and $ν ↔ σ$), and applying anti-symmetry on the $γ$'s and symmetry on the $g$'s, we also get:
$$\begin{align}
γ^{ρσ} γ^{μν}
&= \left(g^{σμ} g^{ρν} - g^{σν} g^{ρμ}\right) + \left(g^{σμ} γ^{ρν} - g^{σν} γ^{ρμ} - g^{ρμ} γ^{σν} + g^{ρν} γ^{σμ}\right) + γ^{ρσμν}\\
&= \left(g^{μσ} g^{νρ} - g^{νσ} g^{μρ}\right) - \left(g^{μσ} γ^{νρ} - g^{νσ} γ^{μρ} - g^{μρ} γ^{νσ} + g^{νρ} γ^{μσ}\right) + γ^{μνρσ}.
\end{align}$$
Finally, subtracting the two, we get the following commutator:
$$\begin{align}
\left[γ^{μν}, γ^{ρσ}\right]
&= γ^{μν} γ^{ρσ} - γ^{ρσ} γ^{μν}\\
&= 2 \left(g^{νρ} γ^{μσ} - g^{νσ} γ^{μρ} - g^{μρ} γ^{νσ} + g^{μσ} γ^{νρ}\right)
\end{align}$$
Finally, dividing by $4$, we get:
$$
\left[\frac{γ^{μν}}2, \frac{γ^{ρσ}}2\right] = g^{νρ} \frac{γ^{μσ}}2 - g^{νσ} \frac{γ^{μρ}}2 - g^{μρ} \frac{γ^{νσ}}2 + g^{μσ} \frac{γ^{νρ}}2.
$$
This could have been done, more directly and cleanly, using vectors $v$, $w$, $x$ and $y$ in place of $γ^μ$, $γ^ν$, $γ^ρ$ and $γ^σ$. You could try your hand at it, to see if you get the result
$$\left[\frac{v∧w}2, \frac{x∧y}2\right] = w·x \frac{v∧y}2 - w·y \frac{v∧x}2 - v·x \frac{w∧y}2 + v·y \frac{w∧x}2.$$
In either form, these are the Lie brackets for $SO(g)$, which is $SO(p,q)$, if the metric $g$ has signature $+^p -^q$. The quantized version of the algebra, with an extra factor of $i ħ$ is obtained by attaching the factor to each generator, defining
$$Σ^{μν} = \frac{i ħ}2 γ^{μν} = \frac{i ħ}2\frac{γ^μ γ^ν - γ^ν γ^μ}2.$$
Then,
$$
\left[Σ^{μν}, Σ^{ρσ}\right] = i ħ \left(g^{νρ} Σ^{μσ} - g^{νσ} Σ^{μρ} - g^{μρ} Σ^{νσ} + g^{μσ} Σ^{νρ}\right).
$$
