Real Clifford algebra for $SO(1,2)$

Question

Is it possible to find 3 2x2 matrices $m_i$ with real coefficients such that their anticommutator $\{m_i,m_j\}=2 \eta_{ij}*1_2$, where $\eta = diag(1,-1,-1)$ and $1_2$ is the $2\times2$ identity matrix? I can of course find them purely imaginary, and suspect there's an obstruction to having them purely real, but cannot prove it.

Answer 1

If the Wikipedia aricle is to be believed, this Clifford algebra is isomorphic to $M_2(\Bbb C)$, and so cannot have a two-dimensional real representations. (It has a unique irreducible representation two-dimensional over the complexes, and so four-dimensional over the reals. )