Google search integration in html form?

I have a html form as search bar on my site like that:

<li class="menubox">
    <form class="search" id="search" method="get" target="_blank" action="">
        <fieldset>
            <input type="text" name="" id="suchbegriff" />
            <button type="submit" name="">Suchen</button>
        </fieldset>
    </form>
</li>

Now I want to integrate the google search function without using their design setup.

<script>
  (function() {
    var cx = '001707752910213856904:e8hcbtzeevm';
    var gcse = document.createElement('script');
    gcse.type = 'text/javascript';
    gcse.async = true;
    gcse.src = (document.location.protocol == 'https:' ? 'https:' : 'http:') +
        '//www.google.com/cse/cse.js?cx=' + cx;
    var s = document.getElementsByTagName('script')[0];
    s.parentNode.insertBefore(gcse, s);
  })();
</script>

Is it possible to bring both together, so I could have 100% control about the design through my css?

With other words, do exists somewhere an action link, input field name and submit field name for the google search?