A way to write the CP3 (complex projective space) is by the following coset (space):
$$\mathbb{C}\mathbb{P}^3=\frac{SU(4)\times SU(2)}{U(3)\times SU(2)}$$
My question is, does the complexification of a product coset is simply the product coset of its complexified factors? I mean
$$\mathbb{C}\mathbb{P}^3_{\mathbb{C}}=\frac{SU(4)_{\mathbb{C}}\times SU(2)_{\mathbb{C}}}{U(3)_{\mathbb{C}}\times SU(2)_{\mathbb{C}}}$$
How can I complexify, for example
$$SU(4)\times SU(2)$$
I know that
$$SU(4)_{\mathbb{C}}\equiv SL(4,\mathbb{C})$$
and
$$SU(2)_{\mathbb{C}}\equiv SL(2,\mathbb{C})$$
So
$$\left(SU(4)\times SU(2)\right)_{\mathbb{C}}\equiv SL(4,\mathbb{C})\times SL(2,\mathbb{C})$$
is true? If not, what it is?