For SO(2n) we can construct the lie algebra elements by using antisymmetric combinations of $\gamma_\mu$ which obey the clifford algebra.

Up to some prefactor the elements $ S_{\mu \nu} = \alpha [\gamma_\mu , \gamma_\nu] $ can be used as generators. Then we can identify the cartan subalgebra with the elements $ H_i = S_{(2i-1)(2i)} $.

Now i would like to use this to find the weights for an element ($A_\mu)$ in the vector representation of SO(2n). For this purpose I used the $\gamma_\mu$ basis and tried to find the weights for $A_\mu \gamma^\mu$.

The problem is that certain elements of the cartan algebra just commute with parts of the $A_\mu \gamma^\mu$ sum. For example:

$H_2 (A_1 \gamma^1) = (2 \alpha \gamma_3 \gamma_4) \cdot (A_1 \gamma^1) = (2 \alpha ) \cdot (A_1 \gamma^1) \gamma_3 \gamma_4 $

since the commutation of the 2 $\gamma$ gives a factor of $(-1)^2$. Raising and lowering indices is without consequence since the metric for the clifford algebra is euclidean ( $\delta ^{\mu \nu}$ ).

But this parts should rather go to zero to get a linear action from $H$ on the vector represention.

What is wrong with the approach above? or should it work? Is there a way to justify that the commutation corresponds to a zero element ( or zero weight if it commutes with the wohle $A_\mu \gamma^\mu$)?