$\DeclareMathOperator\SO{SO}$I'm not asking for a proof but for some hint that might be helpful to understand this "anomaly" in 4 dimensions. I'm aware of the parallelism with the $A_4$ finite group case as shown in the answer by @Benoit Kloeckner which I would also like to see elaborated on as I'm not sure I understood how the non-simplicity arises from the 3 decompositions into pairs of orthogonal planes in The non-simplicity of $SO(4)$ and $A_4$, but I'm more centered on the particularity of 4 dimensions.

$\DeclareMathOperator\SO{SO}$I'm not asking for a proof but for some hint that might be helpful to understand this "anomaly" in 4 dimensions. I'm aware of the parallelism with the $A_4$ finite group case as shown in the answer by @Benoit Kloeckner which I would also like to see elaborated on as I'm not sure I understood how the non-simplicity arises from the 3 decompositions into pairs of orthogonal planes in The non-simplicity of $SO(4)$ and $A_4$, but I'm more centered on the particularity of 4 dimensions.

$\DeclareMathOperator\SO{SO}$I'm not asking for a proof but for some hint that might be helpful to understand this "anomaly" in 4 dimensions. I'm aware of the parallelism with the $A_4$ finite group case as shown in the answer by @Benoit Kloeckner which I would also like to see elaborated on as I'm not sure I understood how the non-simplicity arises from the 3 decompositions into pairs of orthogonal planes (since simple Lie group $\SO(3,1)$ would be a counterexample) in The non-simplicity of $SO(4)$ and $A_4$, but I'm more centered on the particularity of 4 dimensions.

$\DeclareMathOperator\SO{SO}$I'm not asking for a proof but for some hint that might be helpful to understand this "anomaly" in 4 dimensions. I'm aware of the parallelism with the $A_4$ finite group case as shown in the answer by @Benoit Kloeckner which I would also like to see elaborated on as I'm not sure I understood how the non-simplicity arises from the 3 decompositions into pairs of orthogonal planes in The non-simplicity of $SO(4)$ and $A_4$, but I'm more centered on the particularity of 4 dimensions.

I'm not asking for a proof but for some hint that might be helpful to understand this "anomaly" in 4 dimensions. I'm aware of the parallelism with the $A_4$ finite group case as shown in the answer by @Benoit Cloeckner which I would also like to see elaborated on as I'm not sure I understood how the non-simplicity arises from the 3 decompositions into pairs of orthogonal planes (since simple Lie group $SO(3,1)$ would be a counterexample) in The non-simplicity of $SO(4)$ and $A_4$, but I'm more centered on the particularity of 4 dimensions.

$\DeclareMathOperator\SO{SO}$I'm not asking for a proof but for some hint that might be helpful to understand this "anomaly" in 4 dimensions. I'm aware of the parallelism with the $A_4$ finite group case as shown in the answer by @Benoit Kloeckner which I would also like to see elaborated on as I'm not sure I understood how the non-simplicity arises from the 3 decompositions into pairs of orthogonal planes (since simple Lie group $\SO(3,1)$ would be a counterexample) in The non-simplicity of $SO(4)$ and $A_4$, but I'm more centered on the particularity of 4 dimensions.