= Solution
Under $\mathfrak{sl}(2)_\alpha$, a root $x\alpha+y\beta$ has weight $2x-y$. Counting the twelve root spaces and the two-dimensional Cartan subalgebra gives multiplicities one at weights $\pm2$, four at $\pm1$, and four at zero. Therefore
$$
\mathfrak g\downarrow\mathfrak{sl}(2)_\alpha
=d_2\oplus4d_1\oplus3d_0.
$$
Under $\mathfrak{sl}(2)_\beta$, the weight is $-3x+2y$. The multiplicities are two at $\pm3$, one at $\pm2$, two at $\pm1$, and four at zero. Hence
$$
\mathfrak g\downarrow\mathfrak{sl}(2)_\beta
=2d_3\oplus d_2\oplus3d_0.
$$
The dimensions are respectively $3+4\cdot2+3=14$ and $2\cdot4+3+3=14$, verifying both direct-sum decompositions of the <Adjoint representation>.
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