Simultaneous generating sets for flags

We prove that any triple of complete flags in ${\mathbb{R}}^d$ admits a common generating set of size $\lfloor 5d\mathrm{/}3\rfloor$ and that this bound is sharp. This result extends the classical linear-algebraic fact – a consequence of the Bruhat decomposition of ${\text{GL}}_d(\mathbb{R})$ – that any pair of complete flags in ${\mathbb{R}}^d$ admits a common generating set of size $d$. We also deduce an analogue for $m$-tuples of flags with $m>3$.