Over at the $n$-Category Cafe John Baez talks about something called "tetrality." This is where $S_4$ acts on the complex lie algebra $\mathfrak{so}(8,\mathbb{C})$ in a way that factors through the exceptional homomorphism $S_4\to S_3$ and through "triality" $S_3\to\mathrm{Out}(\mathfrak{so}(8,\mathbb{C}))$.
Here's my understanding. We have $\mathfrak{so}(4)\oplus\mathfrak{so}(4)\subset\mathfrak{so}(8)$ and $\mathfrak{so}(4)\cong\mathfrak{so}(3)\oplus\mathfrak{so}(3)$ and complexifying gives $\mathfrak{so}(3,\mathbb{R})\otimes\mathbb{C}\cong\mathfrak{so}(3,\mathbb{C})\cong\mathfrak{sl}(2,\mathbb{C})$ which acts on $\mathbb{C}^2$. If we designate the four copies of $\mathfrak{sl}(2,\mathbb{C})$ within $\mathfrak{so}(8,\mathbb{C})$ by the handles $\mathfrak{sl}(V_i)$ acting on $V_i\cong\mathbb{C}^2$ for $i=1,2,3,4$ then supposedly we may decompose
$$ \mathfrak{so}(8,\mathbb{C}) \cong \big(\,\mathfrak{sl}(V_1)\oplus\mathfrak{sl}(V_2)\oplus\mathfrak{sl}(V_3)\oplus \mathfrak{sl}(V_4)\,\big) \,\oplus\, \big(V_1\otimes V_2\otimes V_3\otimes V_4\big)$$
as vector spaces at least (or as reps of $\bigoplus_{i=1}^4\mathfrak{sl}(V_i)$ I imagine - haven't checked).
Baez (and the original author of the paper he is discussing, Manivel) then say
$$ (V_{\sigma(1)}\otimes V_{\sigma(2)})\oplus(V_{\sigma(3)}\otimes V_{\sigma(4)}) $$
are the three irreps of $\mathfrak{so}(8,\mathbb{C})$, as $\sigma$ ranges over $S_4$. What I don't get is how does $\mathfrak{so}(8,\mathbb{C})$ act on these three vector spaces? A priori I only see them as reps of $\bigoplus_{i=1}^4\mathfrak{sl}(V_i)\subset\mathfrak{so}(8,\mathbb{C})$.