Not mentioned, but perhaps implied, is what world power more than any other has blocked international efforts to punish al-Assad's government: Russia.
'He is lying, and (he) knows he is lying'
Obama hasn't been making the case for military action alone. Kerry has been a key part of that effort not just in diplomatic circles, but also while testifying this week before Congress.
But Putin is not convinced.
The Russian president, in fact, appeared publicly perturbed by at least part of Kerry's arguments. On Wednesday, he said the top U.S. diplomat's answer to Congress regarding the presence of an al Qaeda-linked group in the Syrian opposition "is not nice."
"It was unpleasant for me to see it, because we communicate with them based on the assumption that they are decent people," Putin said of U.S. authorities. "(Kerry) knows he is lying. It's sad."
Diplomatic disagreements are part of the territory, especially when you have countries doing what they can to further their interests. But calling another country's official a liar is something else, and as a result it struck a chord in Washington.
'Such a preposterous comment'
"Sec. Kerry is, as you all know, a decorated combat veteran," State Department spokeswoman Jen Psaki said Thursday, referring to Kerry's military service in the Vietnam War while responding to a question about Putin's remark. "He's had ... more than words aimed at him.
"So he's not losing sleep after such a preposterous comment that was based on an inaccurate quote and was completely mischaracterized."
Some could see that public response from the State Department as a shot across the bow of Putin, the man who -- given his nation's geopolitical prominence and close ties with Syria -- might be seen as key to forging a long-sought diplomatic solution. It and Putin's comment hardly suggests there's much trust between the United States and Russia on Syria or whatever else.
Speaking Thursday to CNN, Russian Press Secretary Dmitry Peskov didn't say that the United States is fabricating evidence or lying, adding that he appreciates "our cooperation with American partners." And, Peskov said, he believes Russia and the United States both want peace in Syria.
"But we disagree with the (idea) that somebody in the world is trying to impose their will on another country to change the regime or government," Peskov said. "It's contrary to the international law."
'Russia continues to hold the council hostage'
Yet, some U.S. officials argue, Russia has been partly responsible for preventing international efforts to do something to halt the Syria bloodshed.
Moscow repeatedly has exercised its veto power on the U.N. Security Council to block resolutions on the crisis, many of which would have targeted its longtime ally.
Samantha Power, the U.S. ambassador to the United Nations, said Thursday, "The system has protected the prerogatives of Russia." Even after what she called "the world's largest chemical weapons attack in a quarter century," the United Nations hasn't been able to act swiftly because of Russia.
"Russia continues to hold the council hostage and shirk its international responsibilities, including as a part to the Chemical Weapons Convention," the U.S ambassador said.
Putin did say this week that he "doesn't exclude" OK'ing a U.N. measure authorizing military action if irrefutable proof is presented showing al-Assad's forces were behind the August 21 attack. Yet he didn't give any indication that he had high expectations of seeing such evidence anytime soon.
'There is no military solution'
Those are the words spoken Thursday night to G20 leaders, including Putin and Obama, by the head of the United Nations.
In prepared remarks posted on the U.N. website, and which Russia's state-run RIA Novosti news agency said he delivered, Secretary General Ban Ki-moon alluded to U.N. efforts collecting and analyzing samples from the site of the recent attack. He also spoke of the dire situation, generally, in Syria and surrounding countries that have taken on its refugees and, in some instances, been affected by spillover violence.
"It is imperative to end this war," Ban said.
Providing weapons to one side or the other isn't the answer, he argued. While not commenting specifically on Obama's call for a military strike, Ban did say the best way to resolve this issue was by talking, not fighting: "There is no military solution."
His answer as to what that solution might be involves, once again, tapping joint U.N.-Arab League envoy Lakhdar Brahimi to help "intensify our efforts towards a return to the negotiating table."