particle physics - The $U(1)$ charge of a representation

My question is about the reduction of a representation of a group $SU(5)$ to irreps of the subgroup $SU(3)\times SU(2) \times U(1)$.

For example the weights of the 10 dimensional representation of SU(5) are

One can identify the irreps of the subgroup by regrouping the dynkin labels into $((a_3 a_4) ,(a_1), a_2)$ such that (denoting $-1$ by $\bar{1}$):

$$(1,1)_{Y} \rightarrow \left\{ \begin{array}{l l} (0 0,0,1 ) \end{array} \right.$$

$$(\overline{3},1)_{Y} \rightarrow \left\{ \begin{array}{l l} (0 1,(0),\bar{1}) \\ (1 \bar{1},(0),\bar{1})\\ (\bar{1}0,(0),0) \end{array} \right.$$

$$(3,2)_{Y} \rightarrow \left\{ \begin{array}{l l} (1 0,1,\bar{1}) \\ (\bar{1} 1,\bar{1},1)\\ (0\bar{1},\bar{1},1)\\ (1 0,\bar{1},0)\\ (\bar{1}1,1,0)\\ (0\bar{1},1,0) \end{array} \right.$$

My problem is: how can I derive the $Y$ charge of the $U(1)$ factor for each of these from the Dynkin labels?

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Edit

The metrictensor for SU(5) is thus

$$G= \frac{1}{5}\left( \begin{array}{cccc} 4 & 3 & 2 & 1 \\ 3 & 6 & 4 & 2 \\ 2 & 4 & 6 & 3 \\ 1 & 2 & 3 & 4 \end{array} \right).$$

However in the reference, Slansky, on page 84 the same exercise is done but the axis have negative values...

$$\tilde{Y}^W = \frac{1}{3} [-2 \;1\, -1\; 2].$$

How come they do not agree?