Use the Array map method to return a modified version of your array:

var newArray = theArray.map(function(v,i,a){
   return v[0].split(",");
});

The function that is passed as the argument to the map method is used to determine the values in the mapped array. As you can see, the function takes each value in the array, splits it by comma, and returns the resulting array of two strings.

The output is then:

[["Split", "each"],["string", "in"],["this", "array"]];

To make this work recursively for arrays of arbitrary depth, you can use:

var newArray = theArray.map(function mapper(v,i,a){
    if(typeof v == "string"){
        return v.split(",");
    } else {
        return v.map(mapper);
    }
});