I Showing that operators follow SU(2) algebra

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For two quantum oscillators, I have raising and lowering operators
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and
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, and the number operator
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. I need to check if operators below follow
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commutation relations.

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Now as far as I know, SU(2) algebra commutation relation is [T_1, T_2] = i ε^ijk T_3. So, should I just get T_1 and T_2 in terms of T_- and T_+ and then try to check if I get they follow the SU(2) commutation relation?
 
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Just a pedantic comment but ##SU(2)## is a group, and ##\mathfrak{su}(2)## is an algebra.
 
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