Question: Let us prove $$\operatorname{Spin}(2n) \supset U(1) \times SU(n)?$$ or disprove it, thus $$\operatorname{Spin}(2n) \not\supset U(1) \times SU(n)?$$
I suggest:
We can prove this is true for $n=1,2,$.
We can disprove this for $n \geq 3$.
We can disprove this for $n=5$.
Are there general principle behind for generic $n$??
For $$n=3, \quad Spin(6)=SU(4) \supset U(1) \times SU(3) $$
For $$n=4, \quad Spin(8) \supset U(1) \times SU(4)? $$ may not be true. For $$n=5, \quad Spin(10) \supset U(1) \times SU(5) ?$$ may not be true