Just a quick reminder that Odin never says Frigga begged for Loki’s life. That is simply how the fandom has chosen to interpret his words. What he said is:
“Frigga is the only reason you are still alive.”
That she advocated for him not to be executed is one possible interpretation of that statement. I’m not saying she didn’t do so. I’m just saying that’s not what he said and I wish people would quit claiming otherwise. Another, equally logical interpretation of his phrasing is, “I know that Frigga is your weakness and that being forced to go on living without her will hurt you more than execution would.”
Alternatively, both could be true. Frigga could have asked for Loki’s life to be spared, and Odin could have acquiesced not out of any loyalty to her but because he realised it would be even more cruel than simply killing Loki.
Any of the above interpretations are valid. I just tire of seeing people reference her advocating for Loki as proof positive of her love for him when we don’t even know that she ever did so. Especially when it is so often used to brush aside or minimize all the ways in which she gaslit and emotionally manipulated him. Even if she had begged for his life, that doesn’t nullify the harm she did to him otherwise by her actions both in his cell in TDW and at Odin’s bedside in Thor (2011).