The J-Householder matrices

Let J=0I-I0∈M2n(C).  Let 0≠u∈C2n be given. A J-Householder matrix corresponding to u is Hu≡I-uuTJ. We show that every symplectic matrix is a product of J-Householder matrices. We present properties of J-Householder matrices, and we also present the possible Jordan Canonical Forms of products of two J-Householder matrices.