  Let $K$ denote a field and let $V$ denote a vector space over $K$ with finite
positive dimension. We consider a pair of linear transformations $A:V \to V$
and $A^*:V \to V$ that satisfy (i)--(iv) below:
  (i) Each of $A$, $A^*$ is diagonalizable.
  (ii) There exists an ordering $V_{0},V_{1},...,V_{d}$ of the eigenspaces of
$A$ such that $A^* V_i \subseteq V_{i-1} + V_{i} + V_{i+1}$ for $0 \leq i \leq
d$, where $V_{-1}=0$, $V_{d+1}=0$.
  (iii) There exists an ordering $V^*_{0},V^*_{1},...,V^*_{\delta}$ of the
eigenspaces of $A^*$ such that $A V^*_i \subseteq V^*_{i-1} + V^*_{i} +
V^*_{i+1}$ for $0 \leq i \leq \delta$, where $V^*_{-1}=0$, $V^*_{\delta+1}=0$.
  (iv) There is no subspace $W$ of $V$ such that both $AW \subseteq W$, $A^* W
\subseteq W$, other than W=0 and $W=V$.
  We call such a pair a tridiagonal pair on $V$. In this note we obtain two
results. First, we show that each of $A,A^*$ is determined up to affine
transformation by the $V_i$ and $V^*_i$. Secondly, we characterize the case in
which the $V_i$ and $V^*_i$ all have dimension one. We prove both results using
a certain decomposition of $V$ called the split decomposition.
