Whether it’s the world’s largest technology company or a small manufacturer trying find its niche, just about every organization looking for a new chief executive faces the same problem: How do you find a leader who can both innovate and create a culture of disciplined execution?

Boards that are searching for CEO candidates often say they’re looking for another Steve Jobs. That’s flawed logic. First, there is no other Steve Jobs. Second, a Steve Jobs clone simply wouldn’t work in most organizations. He was an iconic founder who built a company and a culture that were uniquely suited to his leadership style.

Fortunately, there are plenty of other leaders capable of sustaining both execution and innovation, but boards often don’t know what qualities to seek in a candidate. Such a leader must clearly understand how to pull cultural levers that exploit existing efficiencies, and at the same time must be able to draw out and nurture employees’ ideas. That means encouraging employees to be vulnerable, to take risks, and to share new ideas. So it’s important for the CEO to have what might be called connectivity—the ability to link people with one another and to connect ideas with other ideas. Leaders also need mental agility and—this is crucial—empathy.

Why empathy? Because a large organization has constituents with a wide range of viewpoints. A leader has to not only be able to learn from all these different constituents, but he or she also has to be able to persuade them of his or her ideas. Doing that requires finding out what drives people and where they’re coming from—it requires empathy.

The people-intensive aspect of the CEO job leads a lot of boards to assume that the top candidate must be an extrovert. But I’ve come to see that a complete extrovert would be almost as unsuited to the top job as a complete introvert. What’s needed is an ambivert—someone who can learn to embody both extroverted and introverted tendencies, regardless of natural predisposition.

An ambivert can go out among the masses and motivate people or convey a vision, then retreat to a quiet room and think deeply about strategy or about how to connect ideas into something new that consumers will value. An ambivert can engage with constituents, but can also concentrate. To be an effective leader capable of driving an innovative culture, a CEO has to constantly shuttle between those two poles, which can be psychologically taxing. So an ambivert has to be mentally strong, and good boards must properly assess this resilience ahead of time.

In their search for the leader who is going to take the company to the next level, boards often overlook the question of fit: Is the candidate a good match for the company’s culture and goals?

We’re not saying the next CEO should be a replica of the past CEO. That can be a recipe for disaster, since it destroys authenticity. Corporate cultures often need some shaking up, and the arrival of a new leader is the perfect occasion for that. In order to address the question of cultural fit, the board must determine which pieces of the existing culture should be retained and which are expendable. For example, the company might be willing to let go of some of its bureaucratic rigidity while maintaining a culture of transparency and hard work.

Once the board has figured out which elements of the culture it wants to retain, it should assess whether candidates are truly aligned with those values and capable of carrying them forward. Boards shouldn’t be mesmerized by past accomplishments or a couple of good interviews. The best boards dig deeply, assess properly, and talk to each candidate’s past constituents—bosses, direct reports, board members, clients, customers, regulators, and investors. They don’t just focus on the obvious strengths and admired personality traits; rather, they find people the candidate has upset in the past, and they probe to find out exactly why these people were unhappy.

Contrary to popular belief, some of the best innovators come from within the company. Home-grown candidates often can be among a board’s best options if they’ve been identified early and developed to maximize their innovative potential. That may sound almost heretical coming from a partner in an executive search firm, but it’s true. A lot of boards, unfortunately, still wait too long to start the succession-planning process. They often don’t assess the true innovative potential of their rising stars until it is too late.

In many cases, this shortsightedness creates a vicious circle in which high-potential executives get inadequate exposure to the board and aren’t placed in enough “crucible” leadership-development positions. In the end, that lack of exposure and lack of experience disqualifies them from consideration for the CEO post. If the board handles the succession process correctly, internal candidates can very well be the best cultural fit for the company, well equipped to both innovate and execute.

Fit, potential, and the value of internal candidates are just a few of the elements of what’s perhaps the most important aspect of CEO succession planning: pragmatism. Too often, boards develop unrealistic wish lists and unrealistic expectations. Even if they don’t explicitly ask for “the next Steve Jobs,” they want what amounts to the same thing—a charismatic, visionary, exciting, superhuman CEO whose stream of brilliant ideas will elevate the company to greatness. Wish lists like those, which circumvent tough choices and trade-offs, are why CEO failure rates remain way too high.

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