Example 12 Let $a, b, c, d \in \mathbf{R}^{+}, abcd=1$, and let $T=a(b+c+d)+b(c+d)+cd$.
(1) Find the minimum value of $a^{2}+b^{2}+T$;
(2) Find the minimum value of $a^{2}+b^{2}+c^{2}+T$.
(1) $a^{2}+b^{2}+T=a^{2}+b^{2}+(a+b)(c+d)+a b+c d$
$$\begin{array}{l}
\geqslant 2 a b+2 \sqrt{a b} \cdot 2 \sqrt{c d}+a b+c d \\
=4+3 a b+c d \geqslant 4+2 \cdot \sqrt{3 a b c d} \\
=4+2 \sqrt{3}
\end{array}$$

When $a=b, c=d, 3 a b=c d$, i.e., $a=b=\left(\frac{1}{3}\right)^{\frac{1}{t}}, c=d=3^{\frac{1}{4}}$, the equality holds.
Therefore, the minimum value of $a^{2}+b^{2}+T$ is $4+2 \sqrt{3}$.
(2)
$$\begin{aligned}
a^{2}+b^{2}+c^{2}+T & =a^{2}+b^{2}+c^{2}+(a+b+c) d+a b+b c+c a \\
& \geqslant 3 \sqrt[3]{a^{2} b^{2} c^{2}}+3 \sqrt[3]{a b c} \cdot d+3 \cdot \sqrt[3]{(a b c)^{2}} \\
& =3 \cdot\left[2 \sqrt[3]{(a b c)^{2}}+\sqrt[3]{a b c} \cdot d\right] \\
& \geqslant 3 \cdot 2 \cdot \sqrt{2 \sqrt[3]{(a b c)^{2}} \cdot \sqrt[3]{a b c} \cdot d} \\
& =6 \sqrt{2}
\end{aligned}$$

The equality holds when $a=b=c=\left(\frac{1}{2}\right)^{\frac{1}{4}}, d=2^{\frac{3}{4}}$.
Therefore, the minimum value of $a^{2}+b^{2}+c^{2}+T$ is $6 \sqrt{2}$.