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)?

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 hyperkahler 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 hyperkahler 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)? How does this change when the manifold is quaternionic-Kahler instead of hyperkahler?