We know that
$$SO(n) \times SO(m)\subset SO(n+m) \tag{1}$$
$$\frac{Spin(n)\times Spin(m)}{{\mathbf{Z}/2}}\subset Spin(n+m) \tag{2}$$
are both true.
Eq (1) is true for the obvious reason. Eq (2) can be interpret as mod out the common ${\mathbf{Z}/2}$ center subgroup of $Spin(n)\times Spin(m)$. Then Eq (2) mod 2 gives rise to Eq (1). So this makes sense.
I wanted to prove that whether $${Spin(n)\times Spin(n+2)} \subset Spin(2n+2) \tag{3}$$ may be true for certain cases, or may be false in other cases. Are there general statements that we can prove, for $n\geq 2$?
Here is what I know so far:
$n=m-2=2$, $Spin(2)\times Spin(4) = U(1)\times SU(2)\times SU(2) \subset Spin(6)=SU(4)$ seems true if we choose a proper $U(1)$.
$n=m-2=3$, $Spin(3)\times Spin(5) = SU(2)\times Spin(5) \subset Spin(8)$ seems true. Because $\frac{Spin(3)\times Spin(5)}{{\mathbf{Z}/2}}\subset SO(8)$ is true according to https://math.stackexchange.com/a/4161519/955245, so we double cover both sides and we get $Spin(3)\times Spin(5) \subset Spin(8)$.
Can we keep going to prove $Spin(n)\times Spin(n+2) \subset Spin(2n+2)$ for $n=4$ and so on? Or is there a constraint or counter example which $n$ the equality fails?