BP (NYSE: BP), the U.K.'s second oil supermajor after Royal Dutch Shell (NYSE: RDS-A), reported its second-quarter results this morning. Although the company has bounced back from the lows of last summer, its CEO is facing calls for BP to be broken up to release $100 billion for shareholders.

BP bounces back
Between April and June, BP made a replacement-cost profit of $5.3 billion, below analyst expectations. This compares with a record loss of nearly $17 billion in the second quarter of 2010, as BP fought to control a disastrous oil spill from the Deepwater Horizon rig in the Gulf of Mexico.

Despite this rebound, investors were disappointed by BP's output figures, as production slumped 11% to 3.43 million barrels of oil equivalent a day. This fall is largely due to BP's temporary withdrawal from the Gulf of Mexico, as well as various disposals.

With oil prices rising to $127 in early May, BP emphasized its strong cash flows, which it said would grow "faster than output." Operating cash flow (excluding oil-spill spending) was $9.7 billion in the quarter, resulting in BP's net debt falling below 20% of its market value.

BP is to pay a second-quarter dividend of $0.07 per share on September 20, the same as its first-quarter payout. Alas, this is half its pre-spill level.

BP reshapes itself
In February, BP's CEO, Bob Dudley, warned that, "we expect 2011 to be a year of consolidation as we reset the focus of the company."

Today, he echoed this view, stating, "We expect the momentum of our recovery to build into 2012 and 2013 as new projects come on stream, particularly in higher-margin areas [such as Angola, the North Sea and the Gulf of Mexico]; as we complete current turnaround activity; as we return to work in the Gulf of Mexico; and as uncertainties reduce. At the same time we will increasingly focus both our portfolio and our investments on long-term value growth."

Under Dudley, BP has reached agreements to sell $25 billion of assets -- in Argentina, Colombia, Pakistan, and Vietnam -- as Dudley focuses on exploration, rather than production and refining. Other sales, including the disposal of the Texas City and Carson refineries, should help BP to reach its target of raising $30 billion from selling non-core operations.

After BP took a battering from U.S. politicians, pundits, and analysts, its share price plunged in June 2010. It then rebounded into January of this year.

Since then, BP has lagged both its supermajor rivals and the wider FTSE 100. As I write, its shares are pretty much where they started 2011.

Breaking up BP
Fifteen months ago, BP was fighting for its corporate survival. Today, it faces calls to be broken up in order to unlock its hidden value for its owners.

According to analysts at JPMorgan Cazenove, BP could be worth more than $100 billion more if it followed peers ConocoPhillips (NYSE: COP) and Marathon Oil (NYSE: MRO) by separating its upstream business (exploration and production) from its downstream arm (refining and marketing). They calculate that BP's assets are worth almost three-quarters more than its current share price. 

Other oil majors are trading at a discount to their asset value as well, but in BP's case it trades at a 40% discount to its assets, versus the sector average of 27%.

What next for BP?
Although BP is no longer struggling for its very survival, it has faced a number of knockbacks this year.

For example, its $8 billion Arctic tie-up with Russian rival Rosneft fell apart after legal action by BP's existing Russian partner, Alfa-Access-Renova. Also, BP still faces billions of dollars of fines and damages relating to its Gulf spill.

However, BP recently signed a $7 billion deal with Indian energy giant Reliance Industries for offshore exploration. In addition, it continues to win new licenses across the globe, in areas such as Australia, Azerbaijan, Indonesia, the South China Sea, Trinidad, and the U.K.

For now, there is no immediate prospect of BP carving itself in two, not least because it has yet to settle its Deepwater Horizon litigation and subsequent liabilities. Even so, the possibility of spinning off its refining arm demonstrates the potential value that could be unlocked.

BP shares look far too cheap to me. If BP's dividends start rising once more, or corporate action helps to unlock its hidden value, there could be plenty of upside from here.

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This article has been adapted from our sister site, Fool U.K. The Fool has a disclosure policy.