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By Graham Bowley

Vladimir O. Potanin, a Russian billionaire who made his fortune in banking and natural resources, has been a donor and board member of the Guggenheim Museum since 2002. More recently he gave $6.45 million to the Kennedy Center in Washington, which used some of the money to install the “Russian Lounge,” a meeting space, in the performing arts complex created, in part, by Congress. His name is now inscribed on a wall there.

Though wealthy patrons have long used the arts to advance their individual tastes and social standing, much of the Russian giving is different. While the oligarchs also promote their personal preferences and support a wide range of cultural activities, they often employ philanthropy to celebrate their homeland, depicting it as an enlightened wellspring of masterworks in dance, painting, opera and the like.

The effect, however cultivated, helps burnish the image of a nation whose aggression in Ukraine and election meddling have led it to be viewed by many as a hostile power.

The Russian giving, and the strained relations between the countries, has created something of a minefield for American cultural organizations, many of which depend on philanthropic support and embrace shared aesthetic experiences as opportunities for bridge- building. It presents them with an ethical challenge: are they putting themselves at risk, however unwittingly, of helping to promote a one-sided view of a country that the United States is officially sparring with?

One Russian company employed culture to continue interacting with a high-powered American audience, even after it had been put under sanctions.

The company, VTB, a Russian-government-owned bank under limited sanctions since 2014, held two galas at the Kennedy Center. The first, in October 2016, a month before the American presidential election, featured a special performance by stars of the Bolshoi Ballet. The VTB logo decorated both the stage and the uniforms of the wait staff, and VTB’s president, Andrey Kostin, spoke.

But it also accepted a donation from VTB in 2017. The center said it had recently decided it would no longer accept money from the bank since its president, Mr. Kostin, was placed under full sanctions last year.

“The climate has changed since 2016,” said Ms. Roe.

Surprisingly little attention has been paid to these Russian efforts, even as the Kremlin is accused of using more insidious methods to sway American public opinion and elections. The United States, of course, also employs cultural diplomacy through a program run out of the State Department whose preachy use of the Voice of America during the Cold War is well established. But several experts said the Russian version is more coordinated, more baldly designed to muddy the discussion at a time when that country is perceived by many to be overly aggressive.

Michael R. Carpenter, a former National Security Council adviser to President Obama, said he had noticed years ago how the oligarchs were using cultural philanthropy to stay in contact with influential American political, diplomatic and business leaders.

For decades, the production of art was tightly controlled by the state. Censorship was the norm. The Bolshoi toured, of course, but some of its excursions became threadbare affairs, its programming at times chained to ideological themes.

“Such events,” Mr. Putin said, “are the best and most eloquent way to understand a country that possesses huge humanistic and spiritual potential, a country such as Russia.”

A spokeswoman for VTB, the bank under limited sanctions, said in a statement “that the state or its representatives do not influence VTB’s decisions to sponsor museums, theaters, artistic groups. If we get any requests from state representatives, we review them according to standard procedure.”

Its efforts include the commemoration of a Russian site, Fort Elizabeth, on the island of Kauai, to mark the 200th anniversary of a Russian presence in Hawaii.

Michael McFaul, the American ambassador to Russia from 2012 to 2014, recalled how Mr. Kislyak once told him that he had employed Russian culture as a tool to “get deeper into the fabric of society” in the United States. Mr. McFaul said he made limited efforts to do the same in Russia, once helping to bring through the Chicago Symphony, but never with the kind of resources the oligarchs offered.

The oligarchs resist the idea that their spending advances a national agenda.

The park recreates the 19th-century lifestyle of the Russians who scratched out an existence by farming and fur-trading long before California became a state. Visitors tour the stockade, the Russian Orthodox chapel and a windmill like the one used by the settlers. The signs are in English and Russian, and overhead the flag of the Russian company that once ran the settlement often flies.

“We are working hard not to focus just on the Russian era,” said Sarah Sweedler, who runs the Fort Ross Conservancy, a nonprofit that helps operate the site, “but Russia is the reason for the park, after all.”

It’s certainly the reason Mr. Vekselberg, the oligarch, stepped up at Mr. Kislyak’s request to create a private foundation, funded by his company, to help the park. The Russian president at the time, Dmitri Medvedev, attended the signing of the funding agreement with Mr. Vekselberg and Gov. Arnold Schwarzenegger in California.

Mr. Vekselberg, who is fighting the sanctions, declined to be interviewed.

Last year, agents for the special counsel Robert Mueller stopped Mr. Vekselberg at an airport, checked his electronic devices and sought to question him. Mr. Mueller’s team was interested in Mr. Vekselberg’s contact with Michael Cohen, President Trump’s former lawyer. The two men had had a meeting at Trump Tower in January 2017, just before President Trump’s inauguration. Mr. Vekselberg attended the inauguration with his cousin, Andrew Intrater, an American citizen and major donor to the event.

Mr. Vekselberg has denied being involved in the payments, and said he is only a client of his cousin’s firm. The investigators have not accused either man of wrongdoing.

Mr. Vekselberg’s company was not the venue’s only Russian patron. A few years earlier, the Mikhail Prokhorov Fund, named after the billionaire who then owned the Brooklyn Nets, announced a gift of $1 million to help underwrite an exchange program with the arts organization: “TransCultural Express: American and Russian Arts Today.”