Is the representation of dimension 2 of $SU(2)$ also a representation of the Lorentz group?

Is the representation of dimension 2 of $SU(2)$ also a representation of dimension 2 of the part connected to the identity of the group $SO(3,1)$?

To better explain my question:

The representation of dimension 2 of $SU(2)$ is given by exponentiation of the Pauli matrices:

$ U( pmb{epsilon}) = exp (i pmb{epsilon} cdot pmb{sigma} ) $.

At the same time the Lorentz algebra has generators $J^{pm, i}$ with $i=1, 2, 3$. The commutation relations are:

$[J^{+, i}, J^{+, j} ] = i epsilon^{ijk} J^{+, k}$

$[J^{-, i}, J^{-, j} ] = i epsilon^{ijk} J^{-, k}$

$[J^{+, i}, J^{-, j} ] = 0$.

This means that the algebra $so(3, 1)$ is just the algebra $su(2)$ "twice". Or in symbols: $so(3,1)=su(2) + su(2)$. When building the representation $(frac{1}{2}, 0 )$ one maps $ J^{+, i} $ to the Pauli matrices and the $ J^{-, i} $ to the null matrix, so they contribute nothing to exponential. This way one builds a representation of the part connected to the identity of the Lorentz group by exponentiation:

$Lambda = exp (i pmb{epsilon} cdot pmb{sigma}) $.

This is exactly the fundamental representation of $SU(2)$.

Is my reasoning correct? If not, what did I miss?