We employ lattice Boltzmann simulation to numerically investigate the two-dimensional incompressible flow inside a right-angled isosceles triangular enclosure driven by the tangential motion of its hypotenuse. While the base flow, directly evolved from creeping flow at vanishing Reynolds number, remains stationary and stable for flow regimes beyond $Re\gtrsim 13\,400$, chaotic motion is nevertheless observed from as low as $Re≃ 10\,600$. Chaotic dynamics is shown to arise from the destabilisation, following a variant of the classic Ruelle–Takens route, of a secondary solution branch that emerges at a relatively low $Re≃ 4908$ and appears to bear no connection to the base state. We analyse the bifurcation sequence that takes the flow from steady to periodic and then quasi-periodic and show that the invariant torus is finally destroyed in a period-doubling cascade of a phase-locked limit cycle. As a result, a strange attractor arises that induces chaotic dynamics.