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I need to find the most general matrix X commuting with $ J= D_{g} [J_{2}(2), J_{1}(2), J_{2}(3), J_{1}(3)] $

I also need to find the dimension of $ C(J) $ the centralizer of $ J $.

I have found the general matrix X= $$ \left(\begin{array}{cc|c|cc|c} a & 0 & 0 & 0 & 0 & 0\\ b & a & c & 0 & 0 & 0\\ \hline d & 0 & e & 0 & 0 & 0 \\ \hline 0 & 0 & 0 & f & 0 & 0 \\ 0 & 0 & 0 & g & f & h \\ \hline 0 & 0 & 0 & i & 0 & j\\ \end{array}\right) $$

I am not sure how to find the dimension of C(J).

If I understand what my textbook says: $d= 2 + 1 + 1 + 1 + 2 + 1 + 1 + 1 = 10 $

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You've got it-a basis for the centralizer is the ten different versions of your general commuting matrix in which all but one variable is set to 0 and the other is set to 1. –  Kevin Carlson Aug 9 '12 at 22:15
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