Some of my thoughts here were originally posted in a discussion in the Linked-In Rightshifting group.
In my work on organizational intelligence, I have always tried to make it clear that intelligence of the whole is not equal to the aggregate intelligence of its parts. An organization such as a university can be packed with brainy people, but if they don't talk to each other, the collective intelligence of the whole can be pretty poor.
Something similar applies to emotional intelligence. A large religious organization can be packed with sincere and caring people, mostly with very high personal levels of emotional intelligence, but if they don't trust each other (or conversely trust each other too much), then the collective EQ of the organization could leave much to be desired.
It is also worth noting that intelligence is not always correlated with ethics. Thus some individuals with high personal levels of cognitive and/or emotional intelligence may use their powers for selfish, deceptive or manipulative ends, and this may well have a negative impact on the collective intelligence of the organization.
My primary goal in working with organizations is to increase the collective intelligence of the organization, but I'm very happy if individuals get some personal benefits from this as well. This would apply to emotional intelligence as well as other kinds of intelligence.
In complex systems, the whole is never equal to the sum of its parts. If the whole is greater than the parts, then we have a kind of surplus intelligence. If the whole is less than the parts, then we have a deficit. (I suspect the latter is pretty common.) We may be able to tackle the communication mechanisms and "information systems" (in the broadest possible sense of this phrase) to increase the surplus or reduce the deficit, but because the whole organization is a complex adaptive system then this will change the parts as well, hopefully in positive and productive ways, so the net surplus or deficit might not change, but that's okay as long as both the whole and the parts are better off.
As I see it, organizational intelligence depends on a number of capabilities: information gathering (awareness of relevant events and trends), sense-making, and decision-making, as well as collective memory and communication. I guess emotional intelligence has some of the same elements: emotional awareness (situation awareness, people awareness, self-awareness), a degree of self-discipline, and a high degree of personal and group authenticity. Along with the ability to detect and interpret emotional signals from other people, emotional intelligence brings an awareness of the emotional state of the group, which is not simply an aggregation of the emotional state of individuals. We all know that a group or organization can have an emotional state or mood, which affects all those in and around it. It is a common error to identify the person who expresses this mood most strongly, and to blame this person for causing or amplifying the mood ("scapegoat"), but removing the scapegoat merely shifts the task of expressing the collective mood onto someone else.
Here's where I think emotional intelligence is particularly important. If you want to work with a group or organization in a low emotional state, or with poor morale (and let's face it, that's where a lot of organizations are when we start working with them) you have to be able to engage authentically with the negative emotions of the group without being overwhelmed by the same emotions yourself.