$\frac{Spin(2n-1)\times Spin(2n+1)}{{\mathbf{Z}/2}}\subset SO(4n)?$

Question

Is the following statement true? $$\frac{Spin(2n-1)\times Spin(2n+1)}{{\mathbf{Z}/2}}\subset SO(4n)? \tag{1}$$

• For $n=1$, $SU(2)\subset SO(4)$ true.

• For $n=2$, $\frac{Spin(3)\times Spin(5)}{{\mathbf{Z}/2}}\subset SO(8)$ is true according to https://math.stackexchange.com/a/4161519/955245.

• How about $n \geq 3$ in general?

The eq(1) is motivated by the fact that the center of $Z(Spin(2n+1))=Z(Spin(2n-1))=\mathbf{Z}/2$ and $Z(SO(4n))=\mathbf{Z}/2$.

p.s. In contrast, it is easy to know that $\frac{Spin(n)\times Spin(m)}{{\mathbf{Z}/2}}\subset Spin(n+m)$ and $SO(n) \times SO(m)\subset SO(n+m)$ are both true.