It seems intuitively obvious to me that there cannot be an isomorphism between $\mathrm{SU}(2)$ and $\mathrm{SU}(2)\times\mathrm{SU}(2)$ where SU(2) is the Lie Group with the Pauli matrices as generators and $\mathrm{SU}(2)\times\mathrm{SU}(2)$ is the Direct Product whose generators are formed from the generators of $\mathrm{SU}(2)$ by putting them into $2\times2$ matrices in block diagonal form.
(I was told this is a correct method of taking the Direct Product in an answer to a prior question. See the response "that is one method of taking the Direct Product":
And, if so, is the following argument valid?
$\mathrm{SU}(2)$ is not isomorphic to $\mathrm{SO}(1,3)$ because $\mathrm{SO}(1,3)$ is isomorphic to $\mathrm{SU}(2)\times \mathrm{SU}(2)$. If $\mathrm{SU}(2)$ is isomorphic to $\mathrm{SO}(1,3)$, then $\mathrm{SU}(2)$ is isomorphic to $\mathrm{SU}(2)\times \mathrm{SU}(2)$. But, that is false, therefore $\mathrm{SU}(2)$ cannot be isomorphic to $\mathrm{SO}(1,3)$.