  Let $\Gamma$ denote a $Q$-polynomial distance-regular graph with diameter $D
\geq 3$ and standard module $V$. Recently Ito and Terwilliger introduced four
direct sum decompositions of $V$; we call these the $(\mu,\nu)$--{\it split
decompositions} of $V$, where $\mu, \nu \in \lbrace \downarrow, \uparrow
\rbrace$. In this paper we show that the ($\downarrow,\downarrow$)--split
decomposition and the ($\uparrow,\uparrow$)--split decomposition are dual with
respect to the standard Hermitian form on $V$. We also show that the
($\downarrow,\uparrow$)--split decomposition and the
($\uparrow,\downarrow$)--split decomposition are dual with respect to the
standard Hermitian form on $V$.
