A complex manifold admits an almost complex structure, $J$, which satisfies
$$
{J^i}_j{J^j}_k=-{\delta^i}_k, 
$$
and a Hermitian metric, $g$, which satisfies 
$$
g_{st}{J^s}_i{J^t}_j=g_{ij}. \tag{1}
$$
A quaternionic manifold, on the other hand, admits an almost quaternionic structure which satisfies 
$$
{(J_u)^i}_j{(J_v)^j}_k=-{\delta^i}_k{\delta}_{uv}+\varepsilon_{uvz}{(J_z)^i}_k, 
$$
where $u,v,z=1,2,3$. What is the generalization of (1) for a quaternionic manifold? In other words,
$$\tag{2}
g_{st}{(J_u)^s}_i{(J_v)^t}_j=?
$$
If the metric is Hermitian w.r.t. each complex structure, then the the RHS of (2) must contain $g_{ij}\delta_{uv}$. But are there any other terms on the RHS of (2)?