It has not been a good week for General Electric. Last Friday, the company was the subject of a major story in the New York Times that left many readers shocked that this iconic lion of American business paid no American income taxes last year. In fact, the Times reported, it claimed a tax benefit of $3.2 billion.

The news made news—something you can’t say often about a story on corporate taxes—and really went viral when Jon Stewart picked it up, making the company the subject of his satire. Critics questioned why NBC, which is minority-owned by GE after a major stake was sold to Comcast, didn’t cover the news in its nightly broadcast. (An NBC spokesperson said GE's management had nothing to do with the broadcast ommission and told the Times that its tax efforts were necessary to compete globally and that the company is committed to paying all of the taxes its legally obligated to pay.) And on Wednesday, the liberal groups and Progressives United began calling for GE CEO Jeffrey R. Immelt, whom President Obama named the chair of his council on jobs and competitiveness, to step down from that post.

But as bad as all that might seem for GE, it’s even worse for President Obama. When he first tapped Immelt to chair his council, some groaned that the choice seemed like window dressing for a president badly in need of a more pro-business image makeover. Immelt’s company may be investing in green energy and putting innovative technology to work, but how much time can the CEO of one of the world’s largest companies really spend helping to advise the president on the economy, anyways?

Now, however, the choice seems like it could create more difficulties for Obama. You can’t really blame Immelt for taking the post. If you were the CEO of a major multinational corporation and had the opportunity to advise the president, wouldn’t you take him up on the offer?

But Obama—or at least his staff—should have known that picking a sitting CEO to chair his economic council would send the press scurrying to the company’s financial reports. If a reporter can dig up details about the company’s army of tax officials and tax-focused lobbying, why couldn’t the president’s vetters also envisage the headline risks at hand?

As I’ve written before, if Obama really wanted a representative from business to chair his economic panel, a better choice than the GE chief—or most current CEOs, for that matter—would have been a respected but retired chief executive. Intel founder and Silicon Valley gray beard Andy Grove has been outspoken about what he thinks America needs to do to succeed. Former Procter & Gamble CEO A.G. Lafley may be the most respected CEO of his generation, turning a struggling consumer products giant into an innovation powerhouse.

Jeff Immelt surely has plenty to offer the president as an adviser. His conglomerate’s exposure to so many aspects of the economy puts him in an extremely knowledgeable position as a sounding board. But his current post atop a company that had no tax bill in 2010 has public relations trouble written all over it. Had the president’s staff put a little more imagination to work, they might have seen that.

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