$re = '/\[{2}(?:(?:(?!]{2})[^|])+\|)*((?:(?!]{2})[^|])+)]{2}/';
$str = 'including the [[Royal Danish Academy of Sciences and Letters|Danish Academy of Sciences]], [[Norwegian Academy of Science and Letters|Norwegian Academy of Sciences]], [[Russian Academy of Sciences]], and [[National Academy of Sciences|US National Academy of Sciences]].
including the Danish Academy of Sciences, Norwegian Academy of Sciences, Russian Academy of Sciences, and US National Academy of Sciences.';
$subst = "\1";
$result = preg_replace($re, $subst, $str);
echo "The result of the substitution is ".$result;
Please keep in mind that these code samples are automatically generated and are not guaranteed to work. If you find any syntax errors, feel free to submit a bug report. For a full regex reference for PHP, please visit: http://php.net/manual/en/ref.pcre.php