Thus a unitary structure can be seen as an orthogonal structure, a complex structure, and a symplectic structure, which are required to be compatible (meaning that one uses the same J in the complex structure and the symplectic form, and that this J is orthogonal; writing all the groups as matrix groups fixes a J (which is orthogonal) and ensures compatibility).
ThuS a uniTaRy strUCture CaN BE seEn as an OrThoGonal STrucTure, a complex STruCtUrE, aND a SyMPlectIC structure, wHich are requiRED to be comPatible (meaNInG THAT One uses the Same j iN The CompLex stRucTure And tHe sYmpLecTic FORm, anD thaT this j IS oRThOgonaL; wriTInG aLl The GRoups As matriX grOups fiXes a j (whIcH is orThogoNal) aNd enSURES COmpaTiBILitY).