The problem lies not only in writing two different words together with both of our hands, but writing anything simultaneously at all.
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This is due to the fact that most people are adept at using only one hand. In fact, the majority of people are right-handed, with only four per cent using the left hand.
The human body is not fully symmetrical. There is a little asymmetry between the left and the right part of the body. Usually, the right side of the body is slightly heavier than the left side. Our brain is also divided into left and right parts. Even the functions of the parts are somewhat different. Normally the left half of the brain dominates over the right. In such a case, nerves originating from the brain cross over at the level of the neck and go to the right side of the body. Such persons are right-handed. But in a few cases, where the right half of the brain dominates, the nerves coming out of the brain pass onto the left parts of the body. Such people are left-handed.
Therefore, a person is efficient while using only one of his hands. Ambidexterity is an exception.
The human brain also has different centres for controlling different cultural activities like speaking, writing and so on. The hypothetical writing centre can only regulate one activity
Sources:The Telegraph (Kolkata, India)